# Banach fixed point property for Steinberg groups over commutative rings

Izhar Oppenheim  Department of Mathematics, Ben-Gurion University of the
Negev, Be’er Sheva 84105, Israel

###### Abstract.

The main result of this paper is that all affine isometric actions of higher
rank Steinberg groups over commutative rings on uniformly convex Banach spaces
have a fixed point. We consider Steinberg groups over classical root systems
and our analysis covers almost all such Steinberg groups excluding a single
rank 2 case.

The proof of our main result stems from two independent results - a result
regarding relative fixed point properties of root subgroups of Steinberg
groups and a result regarding passing from relative fixed point properties to
a (global) fixed point property. The latter result is proven in the general
setting of groups graded by root systems and provides a far reaching
generalization of the work of Ershov, Jaikin-Zapirain and Kassabov who proved
a similar result regarding property (T) for such groups.

As an application of our main result, we give new constructions of super-
expanders.

##  1\. Introduction

Given a topological group $\Gamma$ and a Banach space
$\operatorname{\mathbb{E}}$, we say that $\Gamma$ has property
$(F_{\operatorname{\mathbb{E}}})$ if every continuous affine isometric action
of $\Gamma$ on $\operatorname{\mathbb{E}}$ admits a fixed point. For a class
of Banach spaces $\mathcal{E}$, we will say that $\Gamma$ has property
$(F_{\mathcal{E}})$, if $\Gamma$ has property
$(F_{\operatorname{\mathbb{E}}})$ for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$.

The most classical instantiation of property $(F_{\mathcal{E}})$ is property
$(FH)$ which is property $(F_{\mathcal{E}})$ where $\mathcal{E}$ is the class
of all real Hilbert spaces. For a $\sigma$-compact, locally compact group
$\Gamma$, Delorme-Guichardet Theorem states that property (FH) is equivalent
to Kazhdan’s property (T).

In this work, we will consider property $(F_{\mathcal{E}_{uc}})$, where
$\mathcal{E}_{uc}$ is the class of all the uniformly convex Banach spaces. We
recall that a Banach space $\operatorname{\mathbb{E}}$ is called uniformly
convex if there is a function $\delta:(0,2]\rightarrow(0,1]$ that is called
the modulus of convexity such that for every two unit vectors
$\xi,\eta\in\operatorname{\mathbb{E}}$ it holds that if
$\|\xi-\eta\|\geq\varepsilon$, then $\|\frac{\xi+\eta}{2}\|\leq
1-\delta(\varepsilon)$. For example, Hilbert spaces are uniformly convex (due
to the parallelogram equality) and for every $1<p<\infty$, every $L^{p}$ space
is uniformly convex (due to Clarkson’s inequalities).

The study of property $(F_{\mathcal{E}_{uc}})$ was initiated in the seminal
paper of Bader, Furman, Gelander and Monod [BFGM07], in which they proved
(among several other results) that higher rank algebraic groups have property
$(F_{L^{p}})$ for every $L^{p}$ space with $1<p<\infty$. They also conjectured
that higher rank algebraic group should have property
$(F_{\mathcal{E}_{uc}})$. This conjecture can be split into two cases
depending on the local field. For non-Archimedian local fields, the conjecture
of [BFGM07] was settled by the work of V. Lafforgue [Laf09] and the subsequent
work of Liao [Lia14] in which stronger results were proven, i.e., in [Laf09,
Lia14] it was proven that higher rank algebraic groups over non-Archimedian
local fields have strong Banach property (T) which implies property
$(F_{\mathcal{E}_{uc}})$. In the Archimedian case, the conjecture of [BFGM07]
was settled only recently: in [Opp22], the author made a breakthrough and
proved that $\operatorname{SL}_{n}(\mathbb{R}),n\geq 4$ and its lattices have
property $(F_{\mathcal{E}_{uc}})$. The technique of [Opp22] was then
generalized by de Laat and de la Salle [dLdlS23] to yield a proof of property
$(F_{\mathcal{E}_{uc}})$ for every real higher rank algebraic group.

In this paper, we turn our attention to the study of property
$(F_{\mathcal{E}_{uc}})$ for Steinberg groups and elementary Chevalley groups
over commutative rings. For a classical (crystallographic) reduced,
irreducible root system $\Phi$ and a commutative ring $R$, we denote
$\mathbb{G}_{\Phi}(R)$ to be the simply-connected Chevalley group
corresponding to $\Phi$ over $R$. We further denote
$\operatorname{EL}_{\Phi}(R)$ to be elementary Chevalley group corresponding
to $\Phi$ over $R$, i.e., the subgroup of $\mathbb{G}_{\Phi}(R)$ generated by
the root subgroups with respect to the standard torus (we refer the reader to
[EJZK17] and references therein for more detailed definitions). The Steinberg
group $\operatorname{St}_{\Phi}(R)$ is a group extension of
$\operatorname{EL}_{\Phi}(R)$ that “forgets” all the relations of
$\operatorname{EL}_{\Phi}(R)$ apart from the relations of $R$ (as an Abelian
group) in each root subgroup and the Chevalley commutator formula (a more
explicit definition is given in section 2.5 below). For example, in the case
where $\Phi=A_{n}$, the groups discussed above are
$\mathbb{G}_{A_{n}}(R)=\operatorname{SL}_{n+1}(R)$,
$\operatorname{EL}_{A_{n}}(R)=\operatorname{EL}_{n+1}(R)$ and
$\operatorname{St}_{A_{n}}(R)=\operatorname{St}_{n+1}(R)$. With these
notations, our main result is:

######  Theorem 1.1.

Let $\Phi$ be a classical reduced, irreducible root system of rank $\geq 2$
such that $\Phi\neq C_{2}$. For every finitely generated, commutative (unital)
ring $R$, the groups $\operatorname{St}_{\Phi}(R)$ and
$\operatorname{EL}_{\Phi}(R)$ have property $(F_{\mathcal{E}_{uc}})$.

Historically, establishing property (T) (or, equivalently, property (FH)) for
$\operatorname{St}_{\Phi}(R)$ and $\operatorname{EL}_{\Phi}(R)$ was more
challenging than establishing property (T) for higher rank algebraic groups.
Indeed, property (T) for higher rank algebraic groups was proved a few years
after Kazhdan defined property (T). In contrast, property (T) for
$\operatorname{EL}_{n}(R),n\geq 3$ was only established much more recently in
the works of Shalom [Sha06] and Vaserstein [Vas06] using a bounded generation
machinery of Shalom (partial results were proven in [Sha99, KN06]). For
$\operatorname{St}_{n}(R),n\geq 3$, Property (T) was proven by Ershov and
Jaikin-Zapirain [EJZ10] using the machinery of angles between subspaces (which
is very different from the bounded generation machinery of Shalom). This angle
machinery was later generalized by Ershov, Jaikin-Zapirain and Kassabov
[EJZK17] to prove property (T) for $\operatorname{St}_{\Phi}(R)$ and
$\operatorname{EL}_{\Phi}(R)$ for every classical reduced, irreducible root
system $\Phi$.

The proof of our main result has two components: relative property
$(F_{\mathcal{E}_{uc}})$ and synthesis of property $(F_{\mathcal{E}_{uc}})$
(the term “synthesis” is due to Mimura [Mim16]).

###  1.1. Relative property $(F_{\mathcal{E}_{uc}})$

For a topological group $\Gamma$, a subgroup $K<\Gamma$ and a Banach space
$\operatorname{\mathbb{E}}$, we say that the pair $(\Gamma,K)$ has relative
property $(F_{\operatorname{\mathbb{E}}})$ if every continuous, affine,
isometric action of $\Gamma$ on $\operatorname{\mathbb{E}}$ has a $K$-fixed
point. For a class of Banach spaces, we will say that the pair $(\Gamma,K)$
has relative property $(F_{\mathcal{E}})$ if it has relative property
$(F_{\operatorname{\mathbb{E}}})$ for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$.

Our main result regarding relative property $(F_{\mathcal{E}_{uc}})$ is the
following:

######  Theorem 1.2.

Let $\Phi\neq C_{2}$ be a classical, reduced, irreducible root system of rank
$\geq 2$ and $R$ a finitely generated commutative (unital) ring. For
$\alpha\in\Phi$, we denote $K_{\alpha}$ to be the root subgroup of
$\operatorname{St}_{\Phi}(R)$. Then for every $\alpha\in\Phi$, the pair
$(\operatorname{St}_{\Phi}(R),K_{\alpha})$ has relative property
$(F_{\mathcal{E}_{uc}})$.

Most of our work toward proving Theorem 1.2 is proving the relative fix point
property in the case where $\Phi=A_{2}$:

######  Theorem 1.3.

For every $m\in\mathbb{N}$ and every $\alpha\in A_{2}$, the pair
$(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

Our approach for proving Theorem 1.3 is to show that for every
$\operatorname{\mathbb{E}}\in\mathcal{E}_{uc}$ and every
$\xi\in\operatorname{\mathbb{E}}$, there is $\widetilde{\xi}$ in the closure
of the convex hull of the
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$-orbit of $\xi$ such
that $\widetilde{\xi}$ is fixed by $K_{\alpha}$. This is done by defining a
sequence of finitely supported averages of vectors in the orbit of $\xi$ and
showing that this sequence converges to $\widetilde{\xi}$ as above. The method
of proof is very similar to that in [Opp22] and relies of bounding the
differences between averages in the Heisenberg group. However, there is a
difficulty that arises from the fact that here we work with the Heisenberg
group with entries in $\mathbb{Z}[t_{1},...,t_{m}]$, while in [Opp22], the
analysis was for the Heisenberg group with entries in $\mathbb{Z}$. Resolving
this difficulty requires a non-trivial improvement of the methods developed in
[Opp22].

Passing from Theorem 1.3 to Theorem 1.2 when $\Phi$ is simply laced or
$\Phi=F_{4}$ is straight-forward: in those cases every root is an
$A_{2}$-subsystem and thus for every $\alpha\in\Phi$ there is
$\operatorname{St}_{A_{2}}(R)<\operatorname{St}_{\Phi}(R)$ such that
$K_{\alpha}<\operatorname{St}_{A_{2}}(R)$. The cases where
$\Phi=B_{n},C_{n},n\geq 3$ or $\Phi=G_{2}$ require some additional analysis
and are proven via a bounded generation argument.

###  1.2. Synthesis of property $(F_{\mathcal{E}_{uc}})$

After establishing relative fixed point properties for the pairs
$(\operatorname{St}_{\Phi}(R),K_{\alpha}),\alpha\in\Phi$, we need to
“synthesize” the relative fixed point property to a global fixed point
property of the entire group $\operatorname{St}_{\Phi}(R)$. In doing so, we
cannot use bounded generation arguments à la Shalom [Sha06], since the bounded
generation assumptions do not hold for Steinberg groups. In [EJZK17], Ershov,
Jaikin-Zapirain and Kassabov gave a synthesis argument for property (T) for
Steinberg groups that relied on the notion of angle between subspaces.
Although there are some results of the author regarding generalizing the angle
machinery to the Banach setting (see [Opp17a, Opp17b, Opp18]), it seems that
generalizing the approach of [EJZK17] to all uniformly convex Banach spaces is
hard to implement. In [Mim16], Mimura gave a “soft” synthesis argument for the
groups $\operatorname{St}_{A_{n}}(R),n\geq 2$ that does not use bounded
generation or angle arguments and that can be applied to the class of all
uniformly convex Banach spaces. However, the argument in [Mim16] was limited
to $\operatorname{St}_{A_{n}}(R)$.

Our work on synthesis of fixed point properties can be seen as a vast
generalization of the basic idea of Mimura [Mim16]. First, we develop a very
general machinery for synthesis of fixed point properties that is interesting
in its own right:

######  Theorem 1.4.

Let $\Gamma$ be a finitely generated group with a finite Abelianization and
$\mathcal{E}$ a class of uniformly convex Banach spaces such that either is
closed under passing to ultraproducts or $\mathcal{E}=\mathcal{E}_{uc}$. Also,
let $\mathcal{G}$ be a (non-directed) connected graph with a vertex set $V$.
Assume that for every $v\in V$, there are subgroups
$N^{v},H_{+}^{v},H_{-}^{v}<\Gamma$ such that the following holds:

  1. (1)

For every $v\in V$, $N^{v}$ normalizes $H_{+}^{v}$ and $H_{-}^{v}$.

  2. (2)

For every $v\in V$, $\langle H_{+}^{v},H_{-}^{v}\rangle=\Gamma$.

  3. (3)

If $u,v\in V$ such that $u\sim v$, then $H_{\pm}^{u}<\langle
H_{\pm}^{v},N^{v}\rangle,$ and $H_{\pm}^{v}<\langle H_{\pm}^{u},N^{u}\rangle.$

  4. (4)

It holds that

$\langle H_{+}^{v},N^{v}:v\in V\rangle=\Gamma.$

  5. (5)

For every $v\in V$, the pairs $(\Gamma,H_{+}^{v}),(\Gamma,H_{-}^{v})$ have
relative property $(F_{\mathcal{E}})$.

Then $\Gamma$ has property $(F_{\mathcal{E}})$.

Using this machinery, we derive a far reaching generalization of the main
result of [EJZK17] ([EJZK17, Theorem 1.2]):

######  Theorem 1.5.

Let $\Phi$ be a classical reduced irreducible root system of rank $\geq 2$,
$R$ a commutative ring, $\operatorname{St}_{\Phi}(R)$ the Steinberg group and
$\\{K_{\alpha}:\alpha\in\Phi\\}$ the root subgroups of
$\operatorname{St}_{\Phi}(R)$.

If for every $\alpha\in\Phi$, the pair
$(\operatorname{St}_{\Phi}(R),K_{\alpha})$ has relative property
$(F_{\mathcal{E}_{uc}})$, then $\operatorname{St}_{\Phi}(R)$ has property
$(F_{\mathcal{E}_{uc}})$.

We note that Theorem 1.5 is actually a special case of a more general result:
In [EJZK17], Ershov, Jaikin-Zapirain and Kassabov defined the general notion
of a group that is strongly graded over a root system and showed that
Steinberg groups $\operatorname{St}_{\Phi}(R)$ are a special case of this
notion. They then proved a synthesis result for property (T) for groups
strongly graded over a root systems. In Theorem 7.7 below, we generalize their
result and prove a synthesis result in the context of uniformly convex Banach
spaces for groups that are strongly graded over root systems.

###  1.3. Super-expanders

As an application of Theorem 1.1, we derive new constructions of super-
expanders (see exact definition in section 9 below):

######  Theorem 1.6.

Let $n\geq 3,m\in\mathbb{N}$ and let $\\{R_{i}\\}_{i\in\mathbb{N}}$ be a
sequence of finite commutative (unital) rings such that for each $i$, $R_{i}$
is generated by $p_{0}^{(i)}=1,p_{1}^{(i)},...,p_{m}^{(i)}\in R_{i}$ and
$|R_{i}|\rightarrow\infty$. Also, let $\Phi\neq C_{2}$ be a classical reduced
irreducible root system of rank $\geq 2$. Denote
$\phi_{i}:\operatorname{St}_{\Phi}(\mathbb{Z}[t_{1},...,t_{m}])\rightarrow\operatorname{EL}_{\Phi}(R_{i})$
be the epimorphisms induced by the ring epimorphism
$\mathbb{Z}[t_{1},...,t_{m}]\rightarrow R_{i},1\mapsto
r_{0}^{(i)},t_{j}\mapsto r_{j}^{(i)},\forall 1\leq j\leq m$.

For a finite generating set $S$ of
$\operatorname{St}_{\Phi}(\mathbb{Z}[t_{1},...,t_{m}])$ it holds that the
family of Cayley graphs of
$\\{(\operatorname{EL}_{\Phi}(R_{i}),\phi_{i}(S))\\}_{i\in\mathbb{N}}$ is a
super-expander family.

#### Structure of this paper

This paper is organized as follows: In section 2, we cover some needed
preliminaries. In section 3, we prove some bounds on the difference of
averaging operations for the Heisenberg group that are needed for our relative
fixed point result. In section 4, we prove a bound on the word norm growth in
the $A_{2}$ Steinberg group. In section 5, we prove our result regarding
relative fixed point properties of root subgroups in Steinberg group (Theorem
1.2 above). In section 6, we prove Theorem 1.4 (and also some more general
versions). In section 7, we prove a general synthesis result for groups graded
by root systems and derive Theorem 1.5 as a special case. In section 8, we
prove our main result (Theorem 1.1 above). Last, in section 9, we show that
our main result yields new construction of super-expanders (Theorem 1.6).

##  2\. Preliminaries

###  2.1. Basic notation

####  $\lesssim$ notation

We will use the $\lesssim$ notation as follows: For to expressions $X,Y$, we
will write $X\lesssim Y$ if there is a universal constant $C$ such that $X\leq
CY$. We will write $X\lesssim_{B}Y$ if there is a constant $C=C(B)$ such that
$X\leq CY$.

#### Monomials notation

For a monomial $\underline{t}=t_{1}^{n_{1}}...t_{m}^{n_{m}}$ where
$n_{i}\in\mathbb{N}\cup\\{0\\}$, we will denote
$\deg(\underline{t})=n_{1}+...+n_{m}$. We will also use the convention
$t_{1}^{0}...t_{m}^{0}=1$.

#### Graph notation

We will denote (non-directed) graphs by $\mathcal{G}$ and directed graphs by
$\vec{\mathcal{G}}$. For two vertices $u,v$ in $\mathcal{G}$, we will denote
$u\sim v$ if there is an edge between $u$ and $v$ (when there is a chance of
ambiguity regarding $\mathcal{G}$, we will denote $u\sim^{\mathcal{G}}v$). For
two vertices $u,v$ in $\vec{\mathcal{G}}$, we will denote $u\rightarrow v$ if
there is a directed edge from $u$ to $v$.

###  2.2. Linear representations and affine isometric actions of a group

Let $\Gamma$ be a discrete group and $\operatorname{\mathbb{E}}$ a Banach
space. An isometric representation of $\Gamma$ on $\operatorname{\mathbb{E}}$
is a homomorphism $\pi:\Gamma\rightarrow O(\operatorname{\mathbb{E}})$, where
$O(\operatorname{\mathbb{E}})$ denotes the group of all isometric linear
invertible maps from $\operatorname{\mathbb{E}}$ to itself. An affine
isometric action of $\Gamma$ on $\operatorname{\mathbb{E}}$ is a homomorphism
$\rho:G\rightarrow\operatorname{Isom}_{aff}(\operatorname{\mathbb{E}})$, where
$\operatorname{Isom}_{aff}(\operatorname{\mathbb{E}})$ is the group of
bijective affine isometries from $\operatorname{\mathbb{E}}$ to itself. We
recall that $\rho$ is of the form

$\rho(g)\xi=\pi(g)\xi+c(g),\forall\xi\in\operatorname{\mathbb{E}}$

where $\pi:\Gamma\rightarrow O(\operatorname{\mathbb{E}})$ is an isometric
linear representation that is called the linear part of $\rho$ and
$c:\Gamma\rightarrow\operatorname{\mathbb{E}}$ is a $1$-cocycle into $\pi$,
i.e., for every $g,h\in\Gamma$,

$c(gh)=c(g)+\pi(g)c(h).$

Denote $\mathbb{R}[\Gamma]$ to be the group ring of $\Gamma$, i.e., the ring
of functions $f:\Gamma\rightarrow\mathbb{R}$ with finite support where
multiplication is via convolution. Equivalently, an element in
$\mathbb{R}[\Gamma]$ can be represented of as a formal sum $\sum_{g\in
F}a_{g}g$, where $F\subseteq\Gamma$ is a finite set and $\\{a_{g}\\}_{g\in
F}\subseteq\mathbb{R}$. In this convention, the multiplication is defined as

$\left(\sum_{g\in F_{1}}a_{g}g\right)\left(\sum_{g^{\prime}\in
F_{2}}b_{g^{\prime}}g^{\prime}\right)=\sum_{g\in F_{1},g^{\prime}\in
F_{2}}a_{g}b_{g^{\prime}}gg^{\prime}.$

Below, we will use both conventions of $\mathbb{R}[\Gamma]$ according to
convenience.

For an affine isometric action $(\rho,\operatorname{\mathbb{E}})$ and
$f\in\mathbb{R}[\Gamma]$ define

$\rho(f)\xi=\sum_{g\in\Gamma}f(g)\rho(g)\xi,\forall\xi\in\operatorname{\mathbb{E}}$

(note that the sum above is finite and thus well defined). We note that
$\rho(f)$ need not be an isometry, but it is always an affine map, i.e., for
every $f\in\mathbb{R}[\Gamma]$ there is a bounded linear map
$T:\operatorname{\mathbb{E}}\rightarrow\operatorname{\mathbb{E}}$ and a vector
$\xi_{0}\in\operatorname{\mathbb{E}}$ such that for every
$\xi\in\operatorname{\mathbb{E}}$,

$\rho(f)\xi=T\xi+\xi_{0}.$

We also note that since every isometric linear representation
$\pi:\Gamma\rightarrow O(\operatorname{\mathbb{E}})$ is an affine isometric
action, this definition also extends to $\pi(f)$ and in that case $\pi(f)$ is
always a bounded linear map.

We define $\operatorname{Prob}_{c}(\Gamma)\subseteq\mathbb{R}[\Gamma]$ to be
finitely supported probability measures on $\Gamma$, i.e.,
$f\in\operatorname{Prob}_{c}(\Gamma)$ if it is finitely supported, $f(g)\geq
0,\forall g$ and $\sum_{g}f(g)=1$. Equivalently,
$\operatorname{Prob}_{c}(\Gamma)$ is the set of all the formal sums
$\sum_{g\in F}a_{g}g$, where $F\subseteq\Gamma$ is a finite set,
$\\{a_{g}\\}_{g\in F}\subseteq[0,1]$ and $\sum_{g\in F}a_{g}=1$.

######  Observation 2.1.

Let $(\rho,\operatorname{\mathbb{E}})$ be an affine isometric action of
$\Gamma$. For every $g\in\Gamma$, every
$\xi_{1},...,\xi_{n}\in\operatorname{\mathbb{E}}$ and every
$a_{1},...,a_{n}\in[0,1]$ such that $\sum_{i}a_{i}=1$, it holds that

$\rho(g)\left(\sum_{i=1}^{n}a_{i}\xi_{i}\right)=\sum_{i=1}^{n}a_{i}\rho(g)\xi_{i}.$

It follows that for every $f_{1},f_{2}\in\operatorname{Prob}_{c}(\Gamma)$, it
holds that $\rho(f_{1})\rho(f_{2})=\rho(f_{1}f_{2})$.

######  Claim 2.2.

Let $(\rho,\operatorname{\mathbb{E}})$ be an affine isometric action of
$\Gamma$ with a linear part $\pi$. Then for every $f\in\mathbb{R}[\Gamma]$ and
every $\xi_{1},\xi_{2}\in\operatorname{\mathbb{E}}$ it holds that

$\rho(f)\xi_{1}-\rho(f)\xi_{2}=\pi(f)(\xi_{1}-\xi_{2}).$

In particular, for $f\in\operatorname{Prob}_{c}(\Gamma)$,

$\|\rho(f)\xi_{1}-\rho(f)\xi_{2}\|\leq\lVert\xi_{1}-\xi_{2}\|.$

###### Proof.

We note that by the definition of affine map it holds for every $g\in\Gamma$
that

$\rho(g)\xi_{1}-\rho(g)\xi_{2}=\pi(g)(\xi_{1}-\xi_{2}).$

Thus, for every $f\in\mathbb{R}[\Gamma]$ it holds that

$\displaystyle\rho(f)\xi_{1}-\rho(f)\xi_{2}=\sum_{g\in\Gamma}f(g)\rho(g)\xi_{1}-\sum_{g\in\Gamma}f(g)\rho(g)\xi_{2}=$
$\displaystyle\sum_{g\in\Gamma}f(g)(\rho(g)\xi_{1}-\rho(g)\xi_{2})=\sum_{g\in\Gamma}f(g)\pi(g)(\xi_{1}-\xi_{2})=\pi(f)(\xi_{1}-\xi_{2}),$

as needed. ∎

######  Proposition 2.3.

Let $(\rho,\operatorname{\mathbb{E}})$ be an affine isometric action of
$\Gamma$. For every integers $0\leq N\leq n,n>0$, every $g\in\Gamma$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\left\|\rho\left(\frac{1}{n}\sum_{i=0}^{n-1}g^{i}\right)\xi-\rho(g^{N})\rho\left(\frac{1}{n}\sum_{i=0}^{n-1}g^{i}\right)\xi\right\|\leq\frac{N}{n}\|\xi-\rho(g^{n})\xi\|.$

###### Proof.

We note that

$\rho(g^{N})\rho\left(\frac{1}{n}\sum_{i=0}^{n-1}g^{i}\right)=\rho\left(\frac{1}{n}\sum_{i=N}^{n+N-1}g^{i}\right).$

Thus,

$\displaystyle\left\|\rho\left(\frac{1}{n}\sum_{i=0}^{n-1}g^{i}\right)\xi-\rho(g^{N})\rho\left(\frac{1}{n}\sum_{i=0}^{n-1}g^{i}\right)\xi\right\|=\left\|\frac{1}{n}\sum_{i=0}^{n-1}\rho(g^{i})\xi-\frac{1}{n}\sum_{i=N}^{n+N-1}\rho(g^{i})\xi\right\|=$
$\displaystyle\left\|\frac{1}{n}\sum_{i=0}^{N-1}\left(\rho(g^{i})\xi-\rho(g^{i+n})\right)\xi\right\|\leq$
$\displaystyle\frac{1}{n}\sum_{i=0}^{N-1}\left\|\rho(g^{i})\xi-\rho(g^{i})\rho(g^{n})\xi\right\|\leq^{\text{Claim
}\ref{difference to linear claim}}\frac{N}{n}\|\xi-\rho(g^{n})\xi\|,$

as needed. ∎

Let $S$ be a symmetric generating set of $\Gamma$ (we do not assume that $S$
is necessarily finite). For every $g\in\Gamma$, denote $|g|_{S}$ to be the
word norm with respect to $S$, i.e., $|g|_{S}$ is the distance between $g$ and
$e$ in the Cayley graph $\operatorname{Cay}(\Gamma,S)$. For
$f\in\mathbb{R}[\Gamma]$, we denote

$|f|_{S}=\max\\{|g|_{S}:f(g)\neq 0\\}.$

######  Observation 2.4.

Observe that for $f_{1},f_{2}\in\mathbb{R}[\Gamma]$ it holds that
$|f_{1}f_{2}|_{S}\leq|f_{1}|_{S}+|f_{2}|_{S}$.

######  Proposition 2.5.

Let $\operatorname{\mathbb{E}}$ a Banach space, $\Gamma$ a group, $S$ a
symmetric generating set of $\Gamma$ and $\rho$ an affine isometric action of
$\Gamma$ . Assume that

$\sup_{s\in S}\|\rho(s)0\|<\infty.$

Then for every $\xi\in\operatorname{\mathbb{E}}$ and every $g\in\Gamma$ it
holds that

$\|\xi-\rho(g)\xi\|\leq(\sup_{s\in S}\|\rho(s)0\|+2\|\xi\|)|g|_{S}.$

Also, for every $f,f_{1},f_{2}\in\operatorname{Prob}_{c}(\Gamma)$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\|\xi-\rho(f)\xi\|\lesssim_{\|\xi\|,S}|f|_{S},$

and

$\|\rho(f_{1})\xi-\rho(f_{2})\xi\|\lesssim_{\|\xi\|,S}|f_{1}|_{S}+|f_{2}|_{S}.$

###### Proof.

Let $\pi$ denote the linear part of $\rho$ and $c$ denote the $1$-cocycle of
$\rho$. We note that

$\sup_{s\in S}\|c(s)\|=\sup_{s\in S}\|\rho(s)0\|<\infty.$

Fix $\xi\in\operatorname{\mathbb{E}}$ and denote $C=2\|\xi\|+\sup_{s\in
S}\|c(s)\|$. Note that

$\sup_{s\in S}\|\xi-\rho(s)\xi\|=\sup_{s\in S}\|\xi-\pi(s)\xi-c(s)\|\leq
2\|\xi\|+\sup_{s\in S}\|c(s)\|=C.$

We will show that for every $g\in\Gamma$,

$\|\xi-\rho(g)\xi\|\leq C|g|_{S}.$

The proof is by induction on $|g|_{S}$. If $|g|_{S}=0$, then $g=e$ and
$\|\xi-\rho(e)\xi\|=0$ as needed. Assume that the inequality holds for every
$g^{\prime}\in\Gamma$ with $|g^{\prime}|_{S}=n$ and let $g\in\Gamma$ with
$|g|_{S}=n+1$. There are $s_{1}...s_{n+1}\in S$ such that $g=s_{1}...s_{n+1}$.
Denote $g^{\prime}=s_{2}...s_{n+1}$. Then

$\displaystyle\|\xi-\rho(g)\xi\|\leq\|\xi-\rho(s_{1})\xi\|+\|\rho(s_{1})\xi-\rho(s_{1}...s_{n+1})\xi\|\leq$
$\displaystyle
C+\|\rho(s_{1})\xi-\rho(s_{1})\rho(g^{\prime})\xi\|=C+\|\xi-\rho(g^{\prime})\xi\|\leq
C+Cn=C(n+1),$

as needed.

For $f\in\operatorname{Prob}_{c}(\Gamma)$, it holds that

$\displaystyle\|\xi-\rho(f)\xi\|=\left\|\xi-\sum_{g}f(g)\rho(g)\xi\right\|=\left\|\sum_{g}f(g)(\xi-\rho(g)\xi)\right\|\leq$
$\displaystyle\sum_{g}f(g)\|\xi-\rho(g)\xi\|\leq\max_{g,f(g)\neq
0}\|\xi-\rho(g)\xi\|\lesssim_{\|\xi\|,S}|f|_{S}.$

It follows that for $f_{1},f_{2}\in\operatorname{Prob}_{c}(\Gamma)$ it holds
that

$\|\rho(f_{1})\xi-\rho(f_{2})\xi\|\leq\|\rho(f_{1})\xi-\xi\|+\|\xi-\rho(f_{2})\xi\|\lesssim_{\|\xi\|,S}|f_{1}|_{S}+|f_{2}|_{S}.$

∎

###  2.3. Bounded generation and Banach fixed point properties

Let $\Gamma$ be a group with subgroups $H_{1},...,H_{k},\Gamma^{\prime}$. We
say that $H_{1},...,H_{k}$ boundedly generate $\Gamma^{\prime}$ if
$\Gamma^{\prime}\subseteq\langle H_{1},...,H_{k}\rangle$ and
$\sup_{g\in\Gamma^{\prime}}|g|_{H_{1}\cup...\cup H_{k}}<\infty$. Shalom
[Sha99], observed that bounded generation and relative property (T) implies
global property (T). His argument is very simple and requires very little
adaptation to the setting of uniformly convex Banach spaces (this result is
well-known and the proof is given below for completeness):

######  Lemma 2.6 (The bounded generation Lemma).

Let $\Gamma$ be a group with subgroups $H_{1},...,H_{k},\Gamma^{\prime}$ and
$\operatorname{\mathbb{E}}$ be a uniformly convex Banach space. Assume that
$(\Gamma,H_{1}),...,(\Gamma,H_{k})$ have relative property
$(F_{\operatorname{\mathbb{E}}})$ and that $\Gamma^{\prime}\subseteq\langle
H_{1},...,H_{k}\rangle$. If $H_{1},...,H_{k}$ boundedly generate
$\Gamma^{\prime}$, then $(\Gamma,\Gamma^{\prime})$ has relative property
$(F_{\operatorname{\mathbb{E}}})$.

###### Proof.

Let $\rho$ be an affine isometric action of $\Gamma$ on
$\operatorname{\mathbb{E}}$. Denote $S=H_{1}\cup...\cup H_{k}$. By the
assumption that $(\Gamma,H_{1}),...,(\Gamma,H_{k})$ have relative property
$(F_{\operatorname{\mathbb{E}}})$ it follows that for every
$\xi\in\operatorname{\mathbb{E}}$,

$\sup_{s\in S}\|\xi-\rho(s)\xi\|<\infty.$

Thus, by the bounded generation assumption and Proposition 2.5 it follows that
for every $\xi\in\operatorname{\mathbb{E}}$,

$\sup_{g\in\Gamma^{\prime}}\|\xi-\rho(g)\xi\|<\infty,$

i.e., the $\Gamma^{\prime}$ orbit of every $\xi$ is bounded. It follows from
Ryll-Nardzewski Theorem or a circumcenter argument that $\Gamma^{\prime}$ has
property $(F_{\operatorname{\mathbb{E}}})$. ∎

###  2.4. Root systems

Here, we will give the basic definitions and results given in [EJZK17]
regarding root systems.

######  Definition 2.7 (Root system).

Let $E$ be a finite dimensional real vector space. A finite non-empty set
$\Phi\subset E$ is called a root system if

  * •

$\Phi$ spans $E$.

  * •

$\Phi$ does not contain $0$.

  * •

$\Phi$ is closed under inversion, i.e.,
$\alpha\in\Phi\Rightarrow(-\alpha)\in\Phi$.

The vectors in $\Phi$ are called the roots of $\Phi$ and the dimension of $E$
is called the rank of $\Phi$.

######  Definition 2.8.

Let $\Phi$ be a root system in $E$.

  * •

$\Phi$ is called reduced if every $1$-dimensional subspace of $E$ contains at
most two elements in $\Phi$.

  * •

$\Phi$ is called irreducible if there are no disjoint non-empty sets
$\Phi_{1},\Phi_{2}\subset\Phi$ such that $\Phi=\Phi_{1}\cup\Phi_{2}$ and
$\operatorname{span}(\Phi_{1})\cap\operatorname{span}(\Phi_{2})=\\{0\\}$.

  * •

A non-empty set $\Psi\subseteq\Phi$ is called a root subsystem of $\Phi$ if
$\Phi\cap\operatorname{span}(\Psi)=\Psi$.

  * •

$\Phi$ is called regular if every root of $\Phi$ is contained in an
irreducible root subsystem of rank $2$.

We note that these definitions generalize the definition of classical root
systems:

######  Definition 2.9 (Classical root system).

Let $\Phi$ be a root system in a space $E$. $\Phi$ is called a classical root
system if there exists an inner-product $(.,.)$ on $E$ such that for every
$\alpha,\beta\in\Phi$ it holds that

$\frac{2(\alpha,\beta)}{(\beta,\beta)}\in\mathbb{Z}\text{ and
}\alpha-\frac{2(\alpha,\beta)}{(\beta,\beta)}\beta\in\Phi.$

We will give only the basic definitions and facts regarding classical root
systems - a much more complete account can be found in [EJZK17] and references
therein. It is well-known that there is a classification of classical reduced
irreducible root systems. Namely, every reduce irreducible classical root
system is isomorphic to one of the following: $A_{n},B_{n}(n\geq
3),C_{n}(n\geq 2),D_{n}(n\geq 4),F_{4},E_{6},E_{7},E_{8},G_{2}$. A classical
reduced irreducible root system $\Phi$ is called simply laced if all the roots
of $\Phi$ have the same length/norm. We also note that all the classical
reduced irreducible root systems are regular.

######  Definition 2.10.

Let $\Phi$ be a root system in $E$. A linear functional
$f:E\rightarrow\mathbb{R}$ is said to be in general position with respect to
$\Phi$ if

  * •

$f(\alpha)\neq 0$ for every $\alpha\in\Phi$.

  * •

$f(\alpha)\neq f(\beta)$ for every $\alpha,\beta\in\Phi,\alpha\neq\beta$.

We denote $\mathfrak{F}(\Phi)$ to be the set of all the linear functionals
that are general position with respect to $\Phi$.

######  Definition 2.11 (Borel set, boundary, core).

Let $\Phi$ be a root system in $E$ and $f\in\mathfrak{F}(\Phi)$.

  * •

The Borel set of $f$ is the set

$\Phi_{f}=\\{\alpha\in\Phi:f(\alpha)>0\\}.$

(note that it can be that $f\neq f^{\prime}$ and
$\Phi_{f}=\Phi_{f^{\prime}}$).

  * •

We denote $\operatorname{Borel}(\Phi)$ to be the set of all the Borel sets of
$\Phi$ and note that this set is closed under inversion, i.e.,
$\Phi_{1}\in\operatorname{Borel}(\Phi)\Rightarrow-\Phi_{1}\in\operatorname{Borel}(\Phi)$.

  * •

For $\Phi_{1},\Phi_{2}\in\operatorname{Borel}(\Phi)$, $\Phi_{1}$ and
$\Phi_{2}$ are called co-minimal if $\Phi_{1}\cap\Phi_{2}$ spans a
$1$-dimensional space.

  * •

For $\Phi_{1},\Phi_{2}\in\operatorname{Borel}(\Phi)$, $\Phi_{1}$ and
$\Phi_{2}$ are called co-maximal if $\Phi_{1}$ and $-\Phi_{2}$ are co-minimal.

  * •

The boundary of the Borel set $\Phi_{1}$ is the set

$\partial\Phi_{1}=\\{\alpha\in\Phi_{1}:\exists\Phi_{2}\in\operatorname{Borel}(\Phi)\text{
such that }\Phi_{1}\text{ and }\Phi_{2}\text{ are co-minimal and
}\alpha\in\Phi_{1}\cap\Phi_{2}\\}.$

  * •

The core of the Borel set $\Phi_{2}$ is the set

$\operatorname{Core}(\Phi_{1})=\Phi_{1}\setminus\partial\Phi_{1}.$

Below, we will need the following facts from [EJZK17]:

######  Lemma 2.12.

Let $\Phi$ be a root system in $E$.

  1. (1)

[EJZK17, Lemma 4.4] Let $\Phi_{1},\Phi_{2}\in\operatorname{Borel}(\Phi)$ such
that $\Phi_{1}$ and $\Phi_{2}$ are not co-maximal. Then there exists
$\Phi_{3}\in\operatorname{Borel}(\Phi)$ such that
$\Phi_{1}\cap\Phi_{2}\subsetneqq\Phi_{1}\cap\Phi_{3}$ and
$\Phi_{1}\cap\Phi_{2}\subsetneqq\Phi_{2}\cap\Phi_{3}$.

  2. (2)

[EJZK17, Lemma 4.6] Let $\Phi_{1}\in\operatorname{Borel}(\Phi)$ and
$\alpha,\beta\in\Phi_{1}$ such that $\alpha,\beta$ are linearly independent.
For every $a,b\in(0,\infty)$, if $a\alpha+b\beta\in\Phi$, then
$a\alpha+b\beta\in\operatorname{Core}(\Phi_{1})$.

  3. (3)

[EJZK17, Lemma 4.8] Assume that $\Phi$ is regular, then for every
$\alpha\in\Phi$, there is $\Phi_{1}\in\operatorname{Borel}(\Phi)$ such that
$\alpha\in\operatorname{Core}(\Phi_{1})$.

Let $\Phi$ be a root system in $E$. We define $\mathcal{G}_{\text{co-max}}$ to
be the graph with the vertex set $V(\mathcal{G}_{\text{co-
max}})=\operatorname{Borel}(\Phi)$ such that for
$\Phi_{1},\Phi_{2}\in\operatorname{Borel}(\Phi),\Phi_{1}\neq\Phi_{2}$ it holds
that $\Phi_{1}\sim\Phi_{2}$ if and only if $\Phi_{1}$ and $\Phi_{2}$ are co-
maximal.

######  Lemma 2.13.

The graph $\mathcal{G}_{\text{co-max}}$ is connected.

The proof of this lemma is very similar to the proof given for the
connectedness of the small Weyl graph in [EJZK17] and we give the proof for
completeness.

###### Proof.

For $\alpha\in\Phi$, we denote $\mathbb{R}\alpha$ to be the $1$-dimensional
subspace spanned by $\alpha$.

Let $\Phi_{1},\Phi_{2}\in\operatorname{Borel}(\Phi)$, $\Phi_{1}\neq\Phi_{2}$.
We will show that there is a path in $\mathcal{G}_{\text{co-max}}$ connecting
$\Phi_{1}$ and $\Phi_{2}$ by a downward induction on
$|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\cap\Phi_{2}\\}|$. We note that
$|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\cap\Phi_{2}\\}|$ is maximal if and
only if

$|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\cap\Phi_{2}\\}|=|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\\}|-1=|\\{\mathbb{R}\alpha:\alpha\in\Phi_{2}\\}|-1$

and this happens if and only if $\Phi_{1}$ and $\Phi_{2}$ are co-maximal and
thus connected by an edge in $\mathcal{G}_{\text{co-max}}$. Assume that
$\Phi_{1}$ and $\Phi_{2}$ are not co-maximal, then by Lemma 2.12 (1), there is
$\Phi_{3}\in\operatorname{Borel}(\Phi)$ such that
$\Phi_{1}\cap\Phi_{2}\subsetneqq\Phi_{1}\cap\Phi_{3}$ and
$\Phi_{1}\cap\Phi_{2}\subsetneqq\Phi_{2}\cap\Phi_{3}$. Thus

$|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\cap\Phi_{2}\\}|<|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\cap\Phi_{3}\\}|$

and

$|\\{\mathbb{R}\alpha:\alpha\in\Phi_{1}\cap\Phi_{2}\\}|<|\\{\mathbb{R}\alpha:\alpha\in\Phi_{2}\cap\Phi_{3}\\}|.$

It follows from the induction assumption that there is a path in
$\mathcal{G}_{\text{co-max}}$ connecting $\Phi_{1}$ and $\Phi_{3}$ and a path
in $\mathcal{G}_{\text{co-max}}$ connecting $\Phi_{2}$ and $\Phi_{3}$ and thus
there is a path connecting $\Phi_{1}$ and $\Phi_{2}$. ∎

###  2.5. Steinberg groups over rings

Steinberg groups can be thought of as abstract groups defined via the
Chevalley commutator relation. More explicitly, given a classical reduced
irreducible root system $\Phi$ and a ring $R$, the group
$\operatorname{St}_{\Phi}(R)$ is the group generated by the set
$\\{x_{\alpha}(p):\alpha\in\Phi,p\in R\\}$ with the following relations:

  1. (1)

For every $\alpha\in\Phi$ and every $p_{1},p_{2}\in R$,
$x_{\alpha}(p_{1})x_{\alpha}(p_{2})=x_{\alpha}(p_{1}+p_{2})$.

  2. (2)

For every $\alpha,\beta\in\Phi$ and every $p_{1},p_{2}\in R$,

$[x_{\alpha}(p_{1}),x_{\beta}(p_{2})]=\prod_{i,j\in\mathbb{N},i\alpha+j\beta\in\Phi}x_{\alpha+j\beta}(C_{i,j}^{\alpha,\beta},\b{p}_{1}^{i}p_{2}^{j}),$

where the product is taken in order of increasing $i+j$ and
$C_{i,j}^{\alpha,\beta}$ are the constants of the Chevalley commutator
relation for Chevalley groups over fields (see for instance [Car89, Section
5.2]).

The subgroups $K_{\alpha}=\\{x_{\alpha}(p):p\in R\\}$ will be called the root
subgroups of $\operatorname{St}_{\Phi}(R)$.

The most important Steinberg group for our analysis is
$\operatorname{St}_{A_{2}}(R)$, which can be defined explicitly as follows:
For every $p\in R$, we denote

$x_{1,2}(p)=x_{\alpha}(p),x_{2,3}(p)=x_{\beta}(p),x_{1,3}(p)=x_{\alpha+\beta}(p),$
$x_{2,1}(p)=x_{-\alpha}(p),x_{3,2}(p)=x_{-\beta}(p),x_{3,1}(p)=x_{-(\alpha+\beta)}(p).$

With this notation, by [EJZK17, Proposition 7.6], the Steinberg group
$\operatorname{St}_{A_{2}}(R)$ is the group generated by the set
$\\{x_{i,j}(p):1\leq i,j\leq 3,i\neq j,p\in R\\}$ under the relations:

  1. (1)

For every $1\leq i,j\leq 3,i\neq j$ and every $p_{1},p_{2}\in R$,

$x_{i,j}(p_{1})x_{i,j}(p_{2})=x_{i,j}(p_{1}+p_{2}).$

  2. (2)

For every $1\leq i,j,k\leq 3$ that are pairwise distinct and every
$p_{1},p_{2}\in R$,

$[x_{i,j}(p_{1}),x_{j,k}(p_{2})]=x_{i,k}(p_{1}p_{2}).$

  3. (3)

For every $1\leq i,j,k\leq 3$ that are pairwise distinct and every
$p_{1},p_{2}\in R$,

$[x_{i,j}(p_{1}),x_{i,k}(p_{2})]=[x_{j,i}(p_{1}),x_{k,i}(p_{2})]=e.$

We also denote $K_{i,j},i\neq j$ to be the root subgroup

$K_{i,j}=\\{x_{i,j}(p):p\in R\\}.$

We will also need some of the relations for the groups
$\operatorname{St}_{C_{2}}(R)$ and $\operatorname{St}_{G_{2}}(R)$ (these are
not all the relations, but only the relations that are needed for us below):

######  Proposition 2.14.

[EJZK17, Proposition 7.6] For
$\Phi=C_{2}=\\{\pm\alpha,\pm\beta,\pm(\alpha+\beta),\pm(\alpha+2\beta)\\}$
($\alpha$ denotes the long root) it holds for every $p_{1},p_{2}\in R$ that

$[x_{\alpha+\beta}(p_{1}),x_{\beta}(p_{2})]=x_{\alpha+2\beta}(2p_{1}p_{2}),$
$[x_{\alpha}(p_{1}),x_{\beta}(p_{2})]=x_{\alpha+\beta}(p_{1}p_{2})x_{\alpha+2\beta}(p_{1}p_{2}^{2}).$

For
$\Phi=G_{2}=\\{\pm\alpha,\pm\beta,\pm(\alpha+\beta),\pm(\alpha+2\beta),\pm(\alpha+3\beta),\pm(2\alpha+3\beta)\\}$
($\alpha$ denotes the long root) it holds for every $p_{1},p_{2}\in R$ that

$[x_{\alpha}(p_{1}),x_{\beta}(p_{2})]=x_{\alpha+\beta}(p_{1}p_{2})x_{\alpha+2\beta}(p_{1}p_{2}^{2})x_{\alpha+3\beta}(p_{1}p_{2}^{3})x_{2\alpha+3\beta}(p_{1}^{2}p_{2}^{3}),$
$[x_{\alpha}(p_{1}),x_{\alpha+3\beta}(p_{2})]=x_{2\alpha+3\beta}(p_{1}p_{2}).$

##  3\. Action of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on uniformly convex
Banach spaces

Throughout this section, $m\in\mathbb{N}$ is a fixed integer. Below, we will
study the action of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on uniformly
convex Banach spaces and deduce results similar to those of [Opp22] which
dealt with isometric representations of $\rm H_{3}(\mathbb{Z})$.

For every $k\in\mathbb{N}\cup\\{0\\}$ and every
$p\in\mathbb{Z}[t_{1},...,t_{m}]$, we define
$X_{k}(p),Y_{k}(p),Z_{k}(p)\in\operatorname{Prob}_{c}(\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ by

$X_{k}(p)=\frac{e+x(2^{k}p)}{2},Y_{k}(p)=\frac{e+y(2^{k}p)}{2},Z_{k}(p)=\frac{e+z(2^{k}p)}{2}.$

Also, for $d\in\mathbb{N}$, we define

$X^{d}(p)=\frac{1}{2^{d}}\sum_{a=0}^{2^{d}-1}x(ap),Y^{d}=\frac{1}{2^{d}}\sum_{b=0}^{2^{d}-1}y(bp),Z^{d}=\frac{1}{2^{d}}\sum_{c=0}^{2^{d}-1}z(cp).$

We note that

$X^{d}(p)=\prod_{a=0}^{d-1}X_{a}(p),Y^{d}(p)=\prod_{b=0}^{d-1}Y_{b}(p),Z^{d}(p)=\prod_{c=0}^{d-1}Z_{c}(p).$

Let $\underline{t}$ denote monomials of the form
$\underline{t}=t_{1}^{n_{1}}...t_{m}^{n_{m}}$ and recall that we denoted
$\deg(t_{1}^{n_{1}}...t_{m}^{n_{m}})=n_{1}+....+n_{m}$. For
$d\in\mathbb{N}\cup\\{0\\}$, denote $\mathcal{S}^{d}$ to be

$\mathcal{S}^{d}=\\{\underline{t}:\deg(\underline{t})=d\\},$

e.g., $\mathcal{S}^{0}=\\{1\\}$ and $\mathcal{S}^{1}=\\{t_{1},...,t_{m}\\}$.
Also, denote $\mathcal{B}^{d}=\bigcup_{k=0}^{d}\mathcal{S}^{d}$. We observe
that $|\mathcal{S}^{d}|={d+m-1\choose d}$ and $|\mathcal{B}^{d}|={d+m\choose
d}$.

Define

$\underline{X}^{d}=\prod_{\underline{t}\in\mathcal{B}^{d}}X^{d}(\underline{t}),\underline{Y}^{d}=\prod_{\underline{t}\in\mathcal{B}^{d}}Y^{d}(\underline{t}),\underline{Z}^{d}=\prod_{\underline{t}\in\mathcal{B}^{d}}Z^{d}(\underline{t}).$

For $d,d^{\prime}\in\mathbb{N}$, we further define

$\operatorname{Poly}(d,d^{\prime})=\left\\{\sum_{\underline{t}\in\mathcal{B}^{d}}c_{\underline{t}}\underline{t}\in\mathbb{Z}[t_{1},...,t_{m}]:0\leq
c_{\underline{t}}\leq 2^{d^{\prime}}-1\right\\}.$

We note that with this notation,

$\underline{X}^{d}=\frac{1}{|\operatorname{Poly}(d,d)|}\sum_{p\in\operatorname{Poly}(d,d)}x(p),$

and similar equalities hold for $\underline{Y}^{d}$ and $\underline{Z}^{d}$.

The aim of this section is to bound the (norms of) the following differences:
Bounding

(1)
$\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{Y}^{d_{3}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{X}^{d_{1}}\right)\xi\right\|,$

(2)
$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|,$

(3)
$\displaystyle\left\|\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi-\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi\right\|,$

where $\rho$ is an affine isometric action of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on a uniformly convex Banach space
$\operatorname{\mathbb{E}}$.

######  Lemma 3.1.

Let $\operatorname{\mathbb{E}}$ be a Banach space, $\rho$ an affine isometric
action of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on
$\operatorname{\mathbb{E}}$ and
$f,f^{\prime},h_{1},...,h_{k}\in\operatorname{Prob}_{c}(\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ such that $h_{1},...,h_{k}$ are supported
on $\\{z(p):p\in\mathbb{Z}[t_{1},...,t_{m}]\\}$. Assume that there is
$\epsilon\in\mathbb{R}$ such that for every $\xi\in\operatorname{\mathbb{E}}$
it hold that

$\|\rho(f)\xi-\rho(f^{\prime})\xi\|\leq\epsilon\left(\max_{1\leq i\leq
k}\|\xi-\rho(h_{i})\xi\|\right).$

Then for every $f_{1},f_{2}\in\operatorname{Prob}_{c}(\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ it holds that

$\|\rho(f_{1})\rho(f)\rho(f_{2})\xi-\rho(f_{1})\rho(f^{\prime})\rho(f_{2})\xi\|\leq\epsilon\left(\max_{1\leq
i\leq k}\|\xi-\rho(h_{i})\xi\|\right).$

###### Proof.

Let $f_{1},f_{2}\in\operatorname{Prob}_{c}(\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$. We note that by Claim 2.2 it holds that

$\displaystyle\|\rho(f_{1})\rho(f)\rho(f_{2})\xi-\rho(f_{1})\rho(f^{\prime})\rho(f_{2})\xi\|\leq\|\rho(f)\rho(f_{2})\xi-\rho(f^{\prime})\rho(f_{2})\xi\|\leq$
$\displaystyle\epsilon\left(\max_{1\leq i\leq
k}\|\rho(f_{2})\xi-\rho(h_{i})\rho(f_{2})\xi\|\right)=\epsilon\left(\max_{1\leq
i\leq
k}\|\rho(f_{2})\xi-\rho(f_{2})\rho(h_{i})\xi\|\right)\leq\epsilon\left(\max_{1\leq
i\leq k}\|\xi-\rho(h_{i})\xi\|\right).$

∎

######  Lemma 3.2.

Let $d,d^{\prime},d_{2}\in\mathbb{N}$ be constants such that $d\leq d_{2}$ and
$d^{\prime}\leq d_{2}$. Then for every Banach space
$\operatorname{\mathbb{E}}$, every isometric affine action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$, every $\xi\in\operatorname{\mathbb{E}}$
and every $p\in\operatorname{Poly}(d,d^{\prime})$ it holds that

$\left\|\rho(z(p))\rho(\underline{Z}^{d_{2}})\xi-\rho(\underline{Z}^{d_{2}})\xi\right\|\lesssim(d+m)^{m}\left(\frac{1}{2}\right)^{d_{2}-d^{\prime}}\max_{\underline{t}\in\mathcal{B}^{d}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

###### Proof.

Fix $p=\sum_{\underline{t}\in\mathcal{B}^{d}}c_{\underline{t}}\underline{t}$
as above.

We fix an order on $\mathcal{B}^{d}$:

$\mathcal{B}^{d}=\\{\underline{t}(1),\underline{t}(2),...\\}$

and according to this order, we denote the coefficients of $p$ as
$c_{\underline{t}(i)}=c_{i}$.

We denote $\sum_{i=1}^{0}c_{i}\underline{t}(i)=0$ and observe that for every
$\xi$ it holds that

$\left\|\rho(z(p))\rho(\underline{Z}^{d_{2}})\xi-\rho(\underline{Z}^{d_{2}})\xi\right\|\leq\sum_{k=1}^{|\mathcal{B}^{d}|}\left\|\rho\left(z\left(\sum_{i=1}^{k}c_{i}\underline{t}(i)\right)\right)\rho(\underline{Z}^{d_{2}})\xi-\rho\left(z\left(\sum_{i=1}^{k-1}c_{i}\underline{t}(i)\right)\right)\rho(\underline{Z}^{d_{2}})\xi\right\|\leq^{\text{Claim
}\ref{difference to linear
claim}}\sum_{k=1}^{|\mathcal{B}^{d}|}\left\|\rho(z(c_{k}\underline{t}(k)))\rho(\underline{Z}^{d_{2}})\xi-\rho(\underline{Z}^{d_{2}})\xi\right\|.$

We recall that $|\mathcal{B}^{d}|={d+m\choose d}\leq(d+m)^{m}$ and thus it is
enough to show that for every $1\leq c\leq 2^{d^{\prime}}-1$ and every
$\underline{t}\in\mathcal{B}^{d}$ it holds that

$\left\|\rho(z(c\underline{t}))\rho(\underline{Z}^{d_{2}})\xi-\rho(\underline{Z}^{d_{2}})\xi\right\|\lesssim
c\left(\frac{1}{2}\right)^{d_{2}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

Fix $1\leq c\leq 2^{d^{\prime}}-1$ and $\underline{t}\in\mathcal{B}^{d}$. By
Lemma 3.1, it is enough to show that for every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\left\|\rho(z(c\underline{t}))\rho(Z^{d_{2}}(\underline{t}))\xi-\rho(Z^{d_{2}}(\underline{t}))\xi\right\|\lesssim
c\left(\frac{1}{2}\right)^{d_{2}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|,$

and this readily follows from Proposition 2.3. ∎

The following result provides a bound on (1):

######  Theorem 3.3.

Let $d_{1},d_{1}^{\prime},d_{3},d_{3}^{\prime},d_{2}\in\mathbb{N}$ be
constants such that $d_{1}+d_{3}\leq d_{2}$ and
$d_{1}^{\prime}+d_{3}^{\prime}\leq d_{2}$. Also, let
$f,h\in\operatorname{Prob}_{c}(\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ such
that

$\operatorname{supp}(f)\subseteq\\{x(p):p\in\operatorname{Poly}(d_{1},d_{1}^{\prime})\\},$

and

$\operatorname{supp}(h)\subseteq\\{y(p):p\in Poly(d_{3},d_{3}^{\prime})\\}.$

Then for every Banach space $\operatorname{\mathbb{E}}$ and every isometric
affine action $\rho$ of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ on
$\operatorname{\mathbb{E}}$ it holds for every
$\xi\in\operatorname{\mathbb{E}}$ that

$\displaystyle\left\|\rho(fh)\rho(\underline{Z}^{d_{2}})\xi-\rho(hf)\rho(\underline{Z}^{d_{2}})\xi\right\|\lesssim(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{d_{2}-d_{1}^{\prime}-d_{3}^{\prime}}\max_{\underline{t}\in\mathcal{B}^{d_{1}+d_{3}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

In particular, for $d_{1},d_{3},d_{2}\in\mathbb{N}\cup\\{0\\}$, if
$d_{1}+d_{3}\leq d_{2}$, it holds for every Banach space
$\operatorname{\mathbb{E}}$, every isometric affine action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ on $\operatorname{\mathbb{E}}$ and every
$\xi\in\operatorname{\mathbb{E}}$ that

$\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{Y}^{d_{3}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{X}^{d_{1}}\right)\xi\right\|\lesssim\\\
(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{d_{2}-d_{1}-d_{3}}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

###### Proof.

Fix $\rho,\operatorname{\mathbb{E}}$ and $\xi$ as above. Let us denote

$A_{1}=\\{p\in\operatorname{Poly}(d_{1},d_{1}^{\prime}):x(p)\in\operatorname{supp}(f)\\},$
$A_{2}=\\{p\in\operatorname{Poly}(d_{3},d_{3}^{\prime}):y(p)\in\operatorname{supp}(h)\\}.$

We note that

$A_{1}A_{2}\subseteq\operatorname{Poly}(d_{1},d_{1}^{\prime})\operatorname{Poly}(d_{3},d_{3}^{\prime})\subseteq\operatorname{Poly}(d_{1}+d_{3},d_{1}^{\prime}+d_{3}^{\prime}).$

Thus,

$\displaystyle\left\|\rho(fh)\rho(\underline{Z}^{d_{2}})\xi-\rho(hf)\rho(\underline{Z}^{d_{2}})\xi\right\|\leq$
$\displaystyle\sum_{p_{1}\in A_{1},p_{2}\in
A_{2}}f(x(p_{1}))h(y(p_{2}))\left\|\rho(x(p_{1})y(p_{2}))\rho(\underline{Z}^{d_{2}})\xi-\rho(y(p_{2})x(p_{1}))\rho(\underline{Z}^{d_{2}})\xi\right\|=$
$\displaystyle\sum_{p_{1}\in A_{1},p_{2}\in
A_{2}}f(x(p_{1}))h(y(p_{2}))\left\|\rho(y(p_{2})x(p_{1}))\rho(z(p_{1}p_{2}))\rho(\underline{Z}^{d_{2}})\xi-\rho(y(p_{2})x(p_{1}))\rho(\underline{Z}^{d_{2}})\xi\right\|\leq^{\text{Claim
}\ref{difference to linear claim}}$ $\displaystyle\sum_{p_{1}\in
A_{1},p_{2}\in
A_{2}}f(x(p_{1}))h(y(p_{2}))\left\|\rho(z(p_{1}p_{2}))\rho(\underline{Z}^{d_{2}})\xi-\rho(\underline{Z}^{d_{2}})\xi\right\|\lesssim^{\text{Lemma
}\ref{z(p)
lemma},p_{1}p_{2}\in\operatorname{Poly}(d_{1}+d_{3},d_{1}^{\prime}+d_{3}^{\prime})}$
$\displaystyle\sum_{p_{1}\in A_{1},p_{2}\in
A_{2}}f(x(p_{1}))h(y(p_{2}))(d_{1}^{\prime}+d_{3}^{\prime}+m)^{m}\left(\frac{1}{2}\right)^{d_{2}-d_{1}^{\prime}-d_{3}^{\prime}}\max_{\underline{t}\in\mathcal{B}^{d_{1}+d_{3}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|\leq$
$\displaystyle(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{d_{2}-d_{1}^{\prime}-d_{3}^{\prime}}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|,$

as needed. ∎

Bounding (2), (3) will take more work. We start by proving a bound in the case
of the Heisenberg group $\rm H_{3}(\mathbb{Z})$. To ease the reading, in that
case we denote

$X_{k}(1)=X_{k},X^{k}(1)=X^{k},Y_{k}(1)=Y_{k},....,Z^{k}(1)=Z^{k}.$

In [Opp22], the following result was proven:

######  Lemma 3.4.

[Opp22, Lemma 5.4] For every uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$, there is a
constant $r_{1}=r_{1}(\delta)$, $0\leq r_{1}<1$ such that for every isometric
linear representation $\pi:\rm H_{3}(\mathbb{Z})\rightarrow
O(\operatorname{\mathbb{E}})$, every $n,c_{0}\in\mathbb{N}$ such that $1\leq
c_{0}\leq 2^{n-1}$ and every $\zeta\in\operatorname{\mathbb{E}}$, if
$\|I-\pi(z(c_{0}))\zeta\|\geq\frac{1}{2}\|\zeta\|$, then

$\left\|\pi\left(X_{0}Y^{n}\right)\zeta\right\|\leq r_{1}\|\zeta\|.$

Using this Lemma, we prove the following:

######  Theorem 3.5.

For every uniformly convex Banach space $\operatorname{\mathbb{E}}$ with a
modulus of convexity $\delta$, there is a constant $r_{1}=r_{1}(\delta)$,
$0\leq r_{1}<1$ such that for every $n\in\mathbb{N}$, every affine isometric
action $\rho$ of $\rm H_{3}(\mathbb{Z})$ on $\operatorname{\mathbb{E}}$ and
every $\xi\in\operatorname{\mathbb{E}}$ it holds that

$\left\|\rho\left(X_{0}Y^{n}\right)\xi-\rho\left(X_{0}Y^{n}\right)\rho\left(Z_{0}\right)\xi\right\|\leq\max\left\\{r_{1}\|\xi-\rho(Z_{0})\xi\|,\frac{\|\xi-\rho(z(2^{n-1}))\xi\|}{2^{n-1}}\right\\}.$

###### Proof.

Let $\pi$ denote the linear part of $\rho$.

If $\|\xi-\rho(Z_{0})\xi\|\leq\frac{\|\xi-\rho(z(2^{n-1}))\xi\|}{2^{n-1}}$,
then by Claim 2.2,

$\displaystyle\left\|\rho\left(X_{0}Y^{n}\right)\xi-\rho\left(X_{0}Y^{n}\right)\rho\left(Z_{0}\right)\xi\right\|\leq\left\|\xi-\rho\left(Z_{0}\right)\xi\right\|\leq\frac{\|\xi-\rho(z(2^{n-1}))\xi\|}{2^{n-1}},$

and we are done.

Assume that
$\|\xi-\rho(Z_{0})\xi\|>\frac{\|\xi-\rho(z(2^{n-1}))\xi\|}{2^{n-1}}$. Then,
using Claim 2.2,

$\displaystyle\left\|\pi(Z^{n-1})\left(\xi-\rho\left(Z_{0}\right)\xi\right)\right\|=\frac{1}{2}\left\|\pi(Z^{n-1})\left(\xi-\rho\left(z(1)\right)\xi\right)\right\|=$
$\displaystyle\frac{1}{2}\left\|\rho(Z^{n-1})\xi-\rho(Z^{n-1})\rho\left(z(1)\right)\xi\right\|=\frac{1}{2}\left\|\frac{1}{2^{n-1}}\sum_{c=0}^{2^{n-1}-1}\rho(z(c))\xi-\frac{1}{2^{n-1}}\sum_{c=1}^{2^{n-1}}\rho(z(c))\xi\right\|=$
$\displaystyle\frac{1}{2}\left\|\frac{\xi-\rho(z(2^{n-1}))\xi}{2^{n-1}}\right\|<\frac{1}{2}\|\xi-\rho(Z_{0})\xi\|.$

It follows that

$\displaystyle\frac{1}{2^{n-1}}\sum_{c=0}^{2^{n-1}-1}\left\|(I-\pi(z(c)))\left(\xi-\rho\left(Z_{0}\right)\xi\right)\right\|\geq\left\|(I-\pi(Z^{n-1}))\left(\xi-\rho\left(Z_{0}\right)\xi\right)\right\|\geq$
$\displaystyle\left\|\xi-\rho\left(Z_{0}\right)\xi\right\|-\left\|\pi(Z^{n-1})\left(\xi-\rho\left(Z_{0}\right)\xi\right)\right\|>\frac{1}{2}\|\xi-\rho(Z_{0})\xi\|.$

Thus, there is $1\leq c_{0}\leq 2^{n-1}$ such that

$\|(I-\pi(z(c_{0})))\left(\xi-\rho\left(Z_{0}\right)\xi\right)\|\geq\frac{1}{2}\|\xi-\rho(Z_{0})\xi\|,$

and by Lemma 3.4 applied to $\zeta=\xi-\rho(Z_{0})\xi$ it follows that

$\displaystyle\left\|\rho\left(X_{0}Y^{n}\right)\xi-\rho\left(X_{0}Y^{n}\right)\rho\left(Z_{0}\right)\xi\right\|=^{\text{Claim
}\ref{difference to linear
claim}}\left\|\pi\left(X_{0}Y^{n}\right)\left(\xi-\rho\left(Z_{0}\right)\xi\right)\right\|\leq
r_{1}\|\xi-\rho(Z_{0})\xi\|$

as needed. ∎

Going back to the group $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$, we deduce
the following:

######  Corollary 3.6.

Let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$ and
$r_{1}=r_{1}(\delta)$, $0\leq r_{1}<1$ be the constant of Theorem 3.5. For
every $n\in\mathbb{N}$, every $a_{0},b_{0}\in\mathbb{N}\cup\\{0\\}$, every
$p,q\in\mathbb{Z}[t_{1},...,t_{m}]\setminus\\{0\\}$, every affine isometric
action $\rho$ of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on
$\operatorname{\mathbb{E}}$ and every $\xi\in\operatorname{\mathbb{E}}$ it
holds that

$\displaystyle\left\|\rho\left(X_{a_{0}}(p)\left(\prod_{b=0}^{n-1}Y_{b_{0}+b}(q)\right)\right)\xi-\rho\left(X_{a_{0}}(p)\left(\prod_{b=0}^{n-1}Y_{b_{0}+b}(q)\right)\right)\rho\left(Z_{a_{0}+b_{0}}(pq)\right)\xi\right\|\leq$
$\displaystyle\max\left\\{r_{1}\|\xi-\rho(Z_{a_{0}+b_{0}}(pq))\xi\|,\frac{\|\xi-\rho(z(2^{a_{0}+b_{0}+n-1}pq))\xi\|}{2^{n-1}}\right\\}.$

###### Proof.

Fix $\rho,n,a_{0},b_{0},p,q$ as above.

Let $H<H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ be the subgroup $H=\langle
x(2^{a_{0}}p),y(2^{b_{0}}q)\rangle$. We note that $H$ is isomorphic to
$H_{3}(\mathbb{Z})$ via the isomorphism $\phi:H_{3}(\mathbb{Z})\rightarrow H$
induced by $\phi(x(1))=x(2^{a_{0}}p),\phi(y(1))=y(2^{b_{0}}q)$. Note that (by
extending $\phi$ linearly)

$\phi(X_{0})=X_{a_{0}}(p),\phi(Y^{n})=\prod_{b=0}^{n-1}Y_{b_{0}+b}(q).$

Also, note that

$\phi(z(1))=\phi([x(1),y(1)])=[x(2^{a_{0}}p),y(2^{b_{0}}q)]=z(2^{a_{0}+b_{0}}pq),$

and thus $\phi(Z_{0})=Z_{a_{0}+b_{0}}(pq)$.

Define a new isometric affine action $\rho_{0}$ of $H_{3}(\mathbb{Z})$ on
$\operatorname{\mathbb{E}}$ by $\rho_{0}=\rho\circ\phi$. Let
$\xi\in\operatorname{\mathbb{E}}$, then

$\displaystyle\left\|\rho\left(X_{a_{0}}(p)\left(\prod_{b=0}^{n-1}Y_{b_{0}+b}(q)\right)\right)\xi-\rho\left(X_{a_{0}}(p)\left(\prod_{b=0}^{n-1}Y_{b_{0}+b}(q)\right)\right)\rho\left(Z_{a_{0}+b_{0}}(pq)\right)\xi\right\|=$
$\displaystyle\left\|\rho_{0}\left(X_{0}Y^{n}\right)\xi-\rho_{0}\left(X_{0}Y^{n}\right)\rho_{0}\left(Z_{0}\right)\xi\right\|\leq$
$\displaystyle\max\left\\{r_{1}\|\xi-\rho_{0}(Z_{0})\xi\|,\frac{\|\xi-\rho_{0}(z(2^{n-1}))\xi\|}{2^{n-1}}\right\\}=$
$\displaystyle\max\left\\{r_{1}\|\xi-\rho(Z_{a_{0}+b_{0}}(pq))\xi\|,\frac{\|\xi-\rho(z(2^{n-1+a_{0}+b_{0}}pq))\xi\|}{2^{n-1}}\right\\}$

as needed. ∎

As a result, we obtain the following:

######  Lemma 3.7.

Let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$ and
$r_{1}=r_{1}(\delta)$, $0\leq r_{1}<1$ be the constant of Theorem 3.5. Also,
let $n,N\in\mathbb{N}$,
$a_{1},...,a_{N},b_{1},...,b_{N}\in\mathbb{N}\cup\\{0\\}$ and
$p_{1},...,p_{N},q_{1},...,q_{N}\in\mathbb{Z}[t_{1},...,t_{m}]\setminus\\{0\\}$.
Assume there are $d\in\mathbb{N}\cup\\{0\\}$ and
$p\in\mathbb{Z}[t_{1},...,t_{m}]\setminus\\{0\\}$ such that for every $1\leq
k\leq N$ it holds that $a_{k}+b_{k}=d$ and $p_{k}q_{k}=p$. For every $1\leq
k\leq N$, we define $f_{k}\in\operatorname{Prob}_{c}(\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ to be

$f_{k}=X_{a_{k}}(p_{k})\left(\prod_{b=0}^{n-1}Y_{b_{k}+b}(q_{k})\right).$

Then for every affine isometric action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on $\operatorname{\mathbb{E}}$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\prod_{k=1}^{N}f_{k}\right)\xi-\rho\left(\prod_{k=1}^{N}f_{k}\right)\rho\left(Z_{d}(p)\right)\xi\right\|\leq\max\left\\{r_{1}^{N}\|\xi-\rho(Z_{d}(p))\xi\|,\frac{\|\xi-\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}\right\\}.$

In particular, if $N=n$, then

$\displaystyle\left\|\rho\left(\prod_{k=1}^{n}f_{k}\right)\xi-\rho\left(\prod_{k=1}^{n}f_{k}\right)\rho\left(Z_{d}(p)\right)\xi\right\|\leq\max\left\\{r_{1}^{n}\|\xi-\rho(Z_{d}(p))\xi\|,\frac{\|\xi-\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}\right\\}\lesssim$
$\displaystyle\left(\max\left\\{r_{1},\frac{1}{2}\right\\}\right)^{n}\max\left\\{\|\xi-\rho(z(2^{d}p))\xi\|,\|\xi-\rho(z(2^{d+n-1}p))\xi\|\right\\}.$

###### Proof.

Fix $\rho$ and $\xi$ as above. The proof on the inequality is by induction on
$N$. The case $N=1$ was proved in Corollary 3.6.

Assume that the inequality holds for $N$ and let
$a_{1},...,a_{N+1},b_{1},...,b_{N+1}\in\mathbb{N}\cup\\{0\\}$ and
$p_{1},...,p_{N+1},q_{1},...,q_{N+1}\in\mathbb{Z}[t_{1},...,t_{m}]\setminus\\{0\\}$
such that there are $d\in\mathbb{N}\cup\\{0\\}$ and
$p\in\mathbb{Z}[t_{1},...,t_{m}]\setminus\\{0\\}$ such that for every $1\leq
k\leq N+1$ it holds that $a_{k}+b_{k}=d$ and $p_{k}q_{k}=p$. Denote

$f=\prod_{k=2}^{N+1}f_{k}.$

By Corollary 3.6

$\left\|\rho\left(\prod_{k=1}^{N+1}f_{k}\right)\xi-\rho\left(\prod_{k=1}^{N+1}f_{k}\right)\rho\left(Z_{d}(p)\right)\xi\right\|=\left\|\rho\left(f_{1}f\right)\xi-\rho\left(f_{1}f\right)\rho\left(Z_{d}(p)\right)\xi\right\|=\left\|\rho\left(f_{1}\right)\rho(f)\xi-\rho\left(f_{1}\right)\rho\left(Z_{d}(p)\right)\rho(f)\xi\right\|\leq\max\left\\{r_{1}\|\rho(f)\xi-\rho(Z_{d}(p))\rho(f)\xi\|,\frac{\|\rho(f)\xi-\rho(z(2^{d+n-1}p))\rho(f)\xi\|}{2^{n-1}}\right\\}=\max\left\\{r_{1}\|\rho(f)\xi-\rho(f)\rho(Z_{d}(p))\xi\|,\frac{\|\rho(f)\xi-\rho(f)\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}\right\\}.$

If
$r_{1}\|\rho(f)\xi-\rho(f)\rho(Z_{d}(p))\xi\|\leq\frac{\|\rho(f)\xi-\rho(f)\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}$,
we are done by Claim 2.2:

$\frac{\|\rho(f)\xi-\rho(f)\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}\leq\frac{\|\xi-\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}.$

Assume that
$r_{1}\|\rho(f)\xi-\rho(f)\rho(Z_{d}(p))\xi\|>\frac{\|\rho(f)\xi-\rho(f)\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}$,
then by the induction assumption

$\displaystyle
r_{1}\left\|\rho\left(f\right)\xi-\rho\left(f\right)\rho\left(Z_{d}(p)\right)\xi\right\|\leq
r_{1}\max\left\\{r_{1}^{N}\|\xi-\rho(Z_{d}(p))\xi\|,\frac{\|\xi-\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}\right\\}\leq$
$\displaystyle\max\left\\{r_{1}^{N+1}\|\xi-\rho(Z_{d}(p))\xi\|,\frac{\|\xi-\rho(z(2^{d+n-1}p))\xi\|}{2^{n-1}}\right\\}$

as needed. ∎

######  Lemma 3.8.

Let $\operatorname{\mathbb{E}}$ be a Banach space $\operatorname{\mathbb{E}}$
and $d_{2}\in\mathbb{N}$ be a constant. Also, let $n,N\in\mathbb{N},2n<d_{2}$,
$a_{1},...,a_{N},b_{1},...,b_{N}\in\mathbb{N}\cup\\{0\\}$ and
$p_{1},...,p_{N},q_{1},...,q_{N}\in\mathcal{B}^{d_{2}}$ such that for every
$1\leq k\leq N-1$ the following holds:

  * •

$a_{k}\geq a_{k+1}$ and $b_{k}\leq b_{k+1}$

  * •

$\deg(p_{k})\geq\deg(p_{k+1})$ and $\deg(q_{k})\leq\deg(q_{k+1})$

  * •

$a_{k+1}+b_{k}\leq d_{2}-2n$ and $\deg(p_{k+1})+\deg(q_{k})\leq d_{2}$

  * •

Either $a_{k}>a_{k+1}$ or $\deg(p_{k})>\deg(p_{k+1})$

For every $1\leq k\leq N$, we define
$h_{k},f_{k}\in\operatorname{Prob}_{c}(\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}]))$ to be

$h_{k}=\prod_{b=0}^{n-1}Y_{b_{k}+b}(q_{k}),$
$f_{k}=X_{a_{k}}(p_{k})h_{k}=X_{a_{k}}(p_{k})\left(\prod_{b=0}^{n-1}Y_{b_{k}+b}(q_{k})\right).$

Then for every affine isometric action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on $\operatorname{\mathbb{E}}$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\prod_{k=1}^{N}X_{a_{k}}(p_{k})\right)\rho\left(\prod_{k=1}^{N}h_{k}\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi-\rho\left(\prod_{k=1}^{N}f_{k}\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi\right\|\lesssim$
$\displaystyle(N-1)(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

In particular, if $N=n$, then

$\displaystyle\left\|\rho\left(\prod_{k=1}^{n}X_{a_{k}}(p_{k})\right)\rho\left(\prod_{k=1}^{n}h_{k}\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi-\rho\left(\prod_{k=1}^{n}f_{k}\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi\right\|\lesssim$
$\displaystyle(n-1)(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|\lesssim(d_{2}+m)^{m}\left(\frac{3}{4}\right)^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

###### Proof.

By Lemma 3.1 (and the triangle inequality), it is enough to prove that for
every $\rho$, every $\xi$ and every $1\leq k_{0}\leq N-1$ it holds that

$\displaystyle\left\|\rho\left(\prod_{k=k_{0}+1}^{N}X_{a_{k}}(p_{k})\right)\rho\left(h_{k_{0}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi-\rho\left(h_{k_{0}}\right)\rho\left(\prod_{k=k_{0}+1}^{N}X_{a_{k}}(p_{k})\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

Denote $f=\prod_{k=k_{0}+1}^{N}X_{a_{k}}(p_{k})$. We note that by the
assumption that for every $k$, $a_{k}\geq a_{k+1}$ and
$\deg(p_{k})\geq\deg(p_{k+1})$ and at least one of these inequalities is a
sharp. Thus,

$\operatorname{supp}(f)\subseteq\\{x(p):p\in\operatorname{Poly}(\deg(p_{k_{0}+1}),a_{k_{0}+1}+1)\\},$

and

$\operatorname{supp}(h_{k_{0}})\subseteq\\{y(p):p\in\operatorname{Poly}(\deg(q_{k_{0}}),b_{k_{0}}+n)\\}.$

By Theorem 3.3 it holds that

$\displaystyle\left\|\rho\left(f\right)\rho\left(h_{k_{0}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi-\rho\left(h_{k_{0}}\right)\rho\left(f\right)\rho\left(\underline{Z}^{d_{2}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{d_{2}-(a_{k_{0}+1}+1)-(b_{k_{0}}+n)}\max_{\underline{t}\in\mathcal{B}^{\deg(p_{k_{0}+1})+\deg(q_{k_{0}})}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|\lesssim^{a_{k_{0}+1}+b_{k_{0}}\leq
d_{2}-2n}$
$\displaystyle(d_{2}+m)^{m}\left(\frac{1}{2}\right)^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|.$

∎

Combining these two lemmas yields the following result:

######  Theorem 3.9.

Let $d_{1},d_{2},d_{3},n\in\mathbb{N},d^{\prime}\in\mathbb{N}\cup\\{0\\}$ be
constants and
$\underline{t}^{\prime}\in\mathbb{Z}[t_{1},...,t_{m}]\setminus\\{0\\}$ be a
monomial. Also, let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach
space with a modulus of convexity $\delta$. Denote $r_{2}=r_{2}(\delta)<1$ to
be

$r_{2}=\max\left\\{r_{1}(\delta),\frac{3}{4}\right\\},$

where $r_{1}(\delta)$ is the constant of Theorem 3.5.

If there are $n\in\mathbb{N}$, $2n<d_{2}$,
$a_{1},...,a_{n},b_{1},...,b_{n}\in\mathbb{N}\cup\\{0\\}$ and
$p_{1},...,p_{n},q_{1},...,q_{n}\in\mathcal{B}^{d_{2}}$ such that the
following holds:

  * •

For every $1\leq k\leq n$, $a_{k}+b_{k}=d^{\prime}$ and
$p_{k}q_{k}=\underline{t}^{\prime}$

  * •

For every $1\leq k\leq n-1$, $d_{1}-1\geq a_{k}\geq a_{k+1}$ and $b_{k}\leq
b_{k+1}\leq d_{3}-1$

  * •

For every $1\leq k\leq n-1$, $\deg(p_{k})\geq\deg(p_{k+1})$ and
$\deg(q_{k})\leq\deg(q_{k+1})$

  * •

For every $1\leq k\leq n-1$, $a_{k+1}+b_{k}\leq d_{2}-2n$ and
$\deg(p_{k+1})+\deg(q_{k})\leq d_{2}$

  * •

For every $1\leq k\leq n-1$, either $a_{k}>a_{k+1}$ or
$\deg(p_{k})>\deg(p_{k+1})$

Then for every affine isometric action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on $\operatorname{\mathbb{E}}$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{m}r_{2}^{n}\max\left\\{\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|,\|\xi-\rho(z(2^{d^{\prime}}\underline{t}^{\prime}))\xi\|,\|\xi-\rho(z(2^{d^{\prime}+n-1}\underline{t}^{\prime}))\xi\|\right\\}.$

###### Proof.

Fix $\rho$ as above. Denote

$h_{k}=\prod_{b=0}^{n-1}Y_{b_{k}+b}(q_{k}),$
$f_{k}=X_{a_{k}}(p_{k})h_{k}=X_{a_{k}}(p_{k})\left(\prod_{b=0}^{n-1}Y_{b_{k}+b}(q_{k})\right).$

By Lemma 3.1, it is enough to prove that for every
$\xi\in\operatorname{\mathbb{E}}$,

$\left\|\rho\left(\prod_{k=1}^{n}X_{a_{k}}(p_{k})\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\prod_{k=1}^{n}h_{k}\right)\xi-\rho\left(\prod_{k=1}^{n}X_{a_{k}}(p_{k})\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\rho\left(\prod_{k=1}^{n}h_{k}\right)\xi\right\|\lesssim\\\
(d_{2}+m)^{m}r_{2}^{n}\max\left\\{\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|,\|\xi-\rho(z(2^{d^{\prime}}\underline{t}^{\prime}))\xi\|,\|\xi-\rho(z(2^{d^{\prime}+n-1}\underline{t}^{\prime}))\xi\|\right\\}.$

Fix $\xi\in\operatorname{\mathbb{E}}$. Then

$\left\|\rho\left(\prod_{k=1}^{n}X_{a_{k}}(p_{k})\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\prod_{k=1}^{n}h_{k}\right)\xi-\rho\left(\prod_{k=1}^{n}X_{a_{k}}(p_{k})\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\rho\left(\prod_{k=1}^{n}h_{k}\right)\xi\right\|\lesssim^{\text{Lemma
}\ref{change h_k to f_k lemma}}\\\
{(d_{2}+m)^{m}r_{2}^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|}+\left\|\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\prod_{k=1}^{n}f_{k}\right)\xi-\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\prod_{k=1}^{n}f_{k}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\xi\right\|\leq^{\text{Claim
}\ref{difference to linear claim}}\\\
(d_{2}+m)^{m}r_{2}^{n}\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|+\left\|\rho\left(\prod_{k=1}^{n}f_{k}\right)\xi-\rho\left(\prod_{k=1}^{n}f_{k}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\xi\right\|\lesssim^{\text{Lemma
}\ref{a_1,..., lemma}}\\\
(d_{2}+m)^{m}r_{2}^{n}\max\left\\{\max_{\underline{t}\in\mathcal{B}^{d_{2}}}\|\xi-\rho(z(2^{d_{2}}\underline{t}))\xi\|,\|\xi-\rho(z(2^{d^{\prime}}\underline{t}^{\prime}))\xi\|,\|\xi-\rho(z(2^{d^{\prime}+n-1}\underline{t}^{\prime}))\xi\|\right\\}.$

∎

This result has two useful instantiations:

######  Corollary 3.10.

Let $d_{1},d_{2},d_{3}\in\mathbb{N},d_{2}\geq 3$ be constants such that
$d_{1},d_{3}\leq d_{2},d_{1}+d_{3}\geq d_{2}+4$ and
$\operatorname{\mathbb{E}}$ be a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$. Also, let
$r_{2}=r_{2}(\delta)$ be the constant of Theorem 3.9.

Denote $n=\lfloor\frac{\sqrt{d_{1}+d_{3}-d_{2}}}{2}\rfloor$. Then for every
$\underline{t}^{\prime}\in\mathcal{B}^{d_{2}+1}$, every $d_{2}-2n\leq
d^{\prime}\leq d_{2}$, every affine isometric action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on $\operatorname{\mathbb{E}}$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

###### Proof.

Fix $\underline{t}^{\prime},d^{\prime},\rho$ and $\xi$ as above.

The proof is an application of Theorem 3.9. Observe that
$2n\leq\sqrt{2d_{2}}<d_{2}$.

We introduce the following notation: recall that the monomial
$\underline{t}^{\prime}\in\mathcal{B}^{d_{2}+1}$ is of the form
$\underline{t}^{\prime}=t_{1}^{n_{1}}...t_{m}^{n_{m}}$ where
$n_{1},...,n_{m}\in\mathbb{N}\cup\\{0\\}$ and $n_{1}+...+n_{m}\leq d_{2}+1$.
For $k\in\mathbb{Z}$, we denote
$\operatorname{prefix}(\underline{t}^{\prime},k)$ to be the $i$-th prefix of
$\underline{t}^{\prime}$:

$\operatorname{prefix}(\underline{t}^{\prime},k)=\begin{cases}1&k\leq 0\\\
\underline{t}^{\prime}&k>n_{1}+...+n_{m}\\\
t_{1}^{n_{1}}...t_{i}^{n_{i}}t_{i+1}^{k-(n_{1}+...+n_{i})}&n_{1}+...+n_{i}<k\text{
and }n_{1}+...+n_{i+1}\geq k\end{cases}.$

Similarly, for $k\in\mathbb{Z}$, we denote
$\operatorname{suffix}(\underline{t}^{\prime},k)$ to be the $k$-th suffix of
$\underline{t}^{\prime}$, i.e.,

$\operatorname{prefix}(\underline{t}^{\prime},k)=\begin{cases}1&k\leq 0\\\
\underline{t}^{\prime}&k>n_{1}+...+n_{m}\\\
t_{i}^{k-(n_{1}+...+n_{i+1})}t_{i+1}^{n_{i+1}}...t_{m}^{n_{m}}&n_{i+1}+...+n_{m}<k\text{
and }n_{i}+...+n_{m}\geq k\end{cases}.$

We note that for every $k$,

$\operatorname{prefix}(\underline{t}^{\prime},\deg(\underline{t}^{\prime})-k)\operatorname{suffix}(\underline{t}^{\prime},k)=\underline{t}^{\prime}.$

For $1\leq k\leq n$, we define

$a_{k}=d_{1}-2nk,b_{k}=d^{\prime}-a_{k},p_{k}=\operatorname{prefix}(\underline{t}^{\prime},\deg(\underline{t}^{\prime})-k),q_{k}=\operatorname{suffix}(\underline{t}^{\prime},k).$

We will finish the proof by verifying that for this choice, the conditions of
Theorem 3.9 are fulfilled:

  * •

It is obvious that for every $1\leq k\leq n$, $a_{k}+b_{k}=d^{\prime}$ and
$p_{k}q_{k}=\underline{t}^{\prime}$.

  * •

For every $1\leq k\leq n-1$, $d_{1}-1\geq a_{k}>a_{k+1}$ and we note that

$a_{n}=d_{1}-2n^{2}\geq
d_{1}-2\left(\frac{\sqrt{d_{1}+d_{3}-d_{2}}}{2}\right)^{2}={\frac{d_{1}}{2}+\frac{d_{2}}{2}-\frac{d_{3}}{2}\geq^{d_{3}\leq
d_{2}}0},$

i.e., for every $k$, $a_{k}\in\mathbb{N}\cup\\{0\\}$. Also, for every $1\leq
k\leq n-1$, $b_{k}<b_{k+1}$ and we note that

$b_{1}=d^{\prime}-a_{1}=d^{\prime}-d_{1}+2n\geq^{d^{\prime}>d_{2}-2n}d_{2}-d_{1}\geq
0.$

Also,

$\displaystyle b_{n}=d^{\prime}-a_{n}\leq
d^{\prime}-(\frac{d_{1}}{2}+\frac{d_{2}}{2}-\frac{d_{3}}{2})\leq^{d^{\prime}\leq
d_{2}}\frac{d_{3}}{2}+\frac{d_{2}}{2}-\frac{d_{1}}{2}=$ $\displaystyle
d_{3}+\frac{d_{2}-(d_{1}+d_{3})}{2}\leq^{d_{1}+d_{3}\geq d_{2}+4}d_{3}-2,$

i.e., for every $1\leq k\leq n$, $b_{k}\in\mathbb{N}\cup\\{0\\}$ and
$b_{k}\leq d_{3}-1$.

  * •

It is obvious that for every $1\leq k\leq n-1$, $\deg(p_{k})\geq\deg(p_{k+1})$
and $\deg(q_{k})\leq\deg(q_{k+1})$.

  * •

For every $1\leq k\leq n-1$,

$a_{k+1}+b_{k}=d_{1}-2n(k+1)+(d^{\prime}-(d_{1}-2nk))=d^{\prime}+2n\leq
d_{2}+2n.$

Last, we need to show that

$\deg(p_{k+1})+\deg(q_{k})=\deg(\underline{t}^{\prime})\leq d_{2}.$

If $\deg(\underline{t}^{\prime})\leq n$, then

$\deg(p_{k+1})+\deg(q_{k})\leq 2\deg(\underline{t}^{\prime})\leq 2n<d_{2}.$

If $d_{2}+1\geq\deg(\underline{t}^{\prime})>n$, then for every $1\leq k\leq
n$,

$\deg(p_{k+1})+\deg(q_{k})=\deg(\underline{t}^{\prime})-(k+1)+k=\deg(\underline{t}^{\prime})-1\leq
d_{2}.$

∎

######  Corollary 3.11.

Let $d_{1},d_{2},d_{3}\in\mathbb{N},d_{2}\geq 3$ be constants such that
$d_{1},d_{3}\leq d_{2},d_{1}+d_{3}\geq d_{2}+4$ and
$\operatorname{\mathbb{E}}$ be a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$. Also, let
$r_{2}=r_{2}(\delta)$ be the constant of Theorem 3.9.

Denote $n=\lfloor\frac{\sqrt{d_{1}+d_{3}-d_{2}}}{2}\rfloor$. Then for every
$\underline{t}^{\prime}\in\mathcal{S}^{d_{2}+1}$, every ,
$d^{\prime}\in\mathbb{N}\cup\\{0\\},d^{\prime}\leq d_{2}-2n$, every affine
isometric action $\rho$ of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on
$\operatorname{\mathbb{E}}$ and every $\xi\in\operatorname{\mathbb{E}}$ it
holds that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(Z_{d^{\prime}}(\underline{t}^{\prime})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

###### Proof.

Fix $\rho,\xi,\underline{t}^{\prime},d^{\prime}$ as above.

The proof is an application of Theorem 3.9. Observe that
$2n\leq\sqrt{2d_{2}}<d_{2}$.

By the assumption that $d_{1}+d_{3}\geq d_{2}+4$ there are $0\leq a_{0}\leq
d_{1}-1$ and $0\leq b_{0}\leq d_{3}-1$ such that $a_{0}+b_{0}=d^{\prime}$.
Using the notation of $\operatorname{prefix},\operatorname{suffix}$ as in the
proof of the previous corollary, we define

$a_{k}=a_{0},b_{k}=b_{0},p_{k}=\operatorname{prefix}(\underline{t}^{\prime},d_{2}+1-k),q_{k}=\operatorname{suffix}(\underline{t}^{\prime},k).$

Checking that the conditions of Theorem 3.9 are fulfilled is straight-forward
and we leave it for the reader. ∎

Combining these two corollaries leads to the following:

######  Theorem 3.12.

Let $d_{1},d_{2},d_{3}\in\mathbb{N},d_{2}\geq 3$ be constants such that
$d_{1},d_{3}\leq d_{2},d_{1}+d_{3}\geq d_{2}+4$ and
$\operatorname{\mathbb{E}}$ be a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$. Also, let
$r_{2}=r_{2}(\delta)$ be the constant of Theorem 3.9.

Denote $n=\lfloor\frac{\sqrt{d_{1}+d_{3}-d_{2}}}{2}\rfloor$. Then for every
affine isometric action $\rho$ of $\rm H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on
$\operatorname{\mathbb{E}}$ and every $\xi\in\operatorname{\mathbb{E}}$ it
holds that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{2m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

###### Proof.

Fix $\rho,\xi$ as above. We note that

$\underline{Z}^{d_{2}+1}=\underline{Z}^{d_{2}}\left(\prod_{\underline{t^{\prime}}\in\mathcal{B}^{d_{2}}}Z_{d_{2}}(\underline{t}^{\prime})\right)\left(\prod_{\underline{t}^{\prime\prime}\in\mathcal{S}^{d_{2}+1}}\prod_{d^{\prime}=0}^{d_{2}}Z_{d^{\prime}}(\underline{t}^{\prime\prime})\right).$

Denote

$Z_{d_{2}}(\mathcal{B}^{d_{2}})=\left(\prod_{\underline{t^{\prime}}\in\mathcal{B}^{d_{2}}}Z_{d_{2}}(\underline{t}^{\prime})\right),$
$Z^{d_{2}+1}(\mathcal{S}^{d_{2}+1})=\left(\prod_{\underline{t}^{\prime\prime}\in\mathcal{S}^{d_{2}+1}}\prod_{d^{\prime}=0}^{d_{2}}Z_{d^{\prime}}(\underline{t}^{\prime\prime})\right).$

Thus,

$\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\leq\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|+\\\
\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})Z^{d_{2}+1}(\mathcal{S}^{d_{2}+1})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|=\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|+\\\
\left\|\rho\left(Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z^{d_{2}+1}(\mathcal{S}^{d_{2}+1})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\leq^{\text{Claim
}\ref{difference to linear
claim}}\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|+\\\
\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z^{d_{2}+1}(\mathcal{S}^{d_{2}+1})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|.$

It follows that it is enough to prove that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{2m+1}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|$

and

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z^{d_{2}+1}(\mathcal{S}^{d_{2}+1})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{2m+1}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

We observe that $|\mathcal{B}^{d_{2}}|={d_{2}+m\choose d_{2}}$. Fix an order
on $\mathcal{B}^{d_{2}}$:

$\mathcal{B}^{d_{2}}=\\{\underline{t}^{\prime}(1),\underline{t}^{\prime}(2),...\\}.$

Thus

$\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\mathcal{B}^{d_{2}})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\leq\sum_{k=0}^{|\mathcal{B}^{d_{2}}|-1}\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\left(\prod_{i=1}^{k}Z_{d_{2}}(\underline{t}^{\prime}(i))\right)\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\left(\prod_{i=1}^{k+1}Z_{d_{2}}(\underline{t}^{\prime}(i))\right)\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|=\sum_{k=0}^{|\mathcal{B}^{d_{2}}|-1}\left\|\rho\left(\prod_{i=1}^{k}Z_{d_{2}}(\underline{t}^{\prime}(i))\right)\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\\\
-\rho\left(\prod_{i=1}^{k}Z_{d_{2}}(\underline{t}^{\prime}(i))\right)\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\underline{t}^{\prime}(k+1))\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\leq^{\text{Claim
}\ref{difference to linear
claim}}\sum_{k=0}^{|\mathcal{B}^{d_{2}}|-1}\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d_{2}}(\underline{t}^{\prime}(k+1))\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim^{\text{Corollary
}\ref{d' geq d_2 -2n coro}}\\\
\sum_{k=0}^{|\mathcal{B}^{d_{2}}|-1}(d_{2}+m)^{m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|\lesssim\\\
(d_{2}+m)^{2m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|,$

as needed.

The proof of the second inequality is similar and we allow ourselves to omit
some details. We recall that $|\mathcal{S}^{d_{2}+1}|={d_{2}+m\choose
d_{2}+1}$ and fix an order on $\mathcal{S}^{d_{2}+1}$:

$\mathcal{S}^{d_{2}+1}=\\{\underline{t}^{\prime\prime}(1),\underline{t}^{\prime\prime}(2),...\\}.$

Thus,

$\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z^{d_{2}+1}(\mathcal{S}^{d_{2}+1})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\leq\sum_{k=0}^{|\mathcal{S}^{d_{2}+1}|-1}\sum_{d^{\prime}=0}^{d_{2}}\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}Z_{d^{\prime}}(\underline{t}^{\prime\prime}(k+1))(\mathcal{S}^{d_{2}+1})\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\\\
\lesssim^{\text{Corollaries }\ref{d' geq d_2 -2n coro},\ref{d' < d_2 -2n
coro}}\sum_{k=0}^{|\mathcal{S}^{d_{2}+1}|-1}\sum_{d^{\prime}=0}^{d_{2}}(d_{2}+m)^{m}r_{2}^{n}\max_{0\leq
d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|\\\
\lesssim(d_{2}+m)^{2m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|,$

as needed. ∎

As a consequence, we can bound (2) and (3):

######  Theorem 3.13.

Let $d_{1},d_{2},d_{3}\in\mathbb{N},d_{2}\geq 16$ be constants such that
$\min\\{d_{1},d_{3}\\}\geq 16$, $\max\\{d_{1},d_{3}\\}\geq d_{2}$ and
$\operatorname{\mathbb{E}}$ be a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity $\delta$. Also, let
$r=r(\delta)=\sqrt[4]{r_{2}(\delta)}<1$ where $r_{2}(\delta)$ is the constant
of Theorem 3.9.

For every affine isometric action $\rho$ of $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ on $\operatorname{\mathbb{E}}$ and every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi-\rho\left(\underline{X}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{Y}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{2m}r^{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}\max_{0\leq
d\leq
2d_{2}}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|,$

and

$\displaystyle\left\|\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi-\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{2m}r^{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}\max_{0\leq
d\leq
2d_{2}}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

###### Proof.

We will start by proving the first inequality. Fix $\rho$ as above.

We denote
$d_{1}^{\prime}=\min\\{d_{1},d_{2}\\},d_{3}^{\prime}=\min\\{d_{3},d_{2}\\}$.
By Lemma 3.1, it is enough to prove that for every
$\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\underline{X}^{d_{1}^{\prime}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}^{\prime}}\right)\xi-\rho\left(\underline{X}^{d_{1}^{\prime}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{Y}^{d_{3}^{\prime}}\right)\xi\right\|\lesssim$
$\displaystyle(d_{2}+m)^{2m}r^{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}\max_{0\leq
d\leq
2d_{2}}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

Note that $d_{1}^{\prime}+d_{3}^{\prime}\geq
d_{2}+\min\\{d_{1},d_{3},d_{2}\\}\geq d_{2}+4$ and thus we can apply Theorem
3.12. Namely, we denote
$n=\lfloor\frac{\sqrt{d_{1}^{\prime}+d_{3}^{\prime}-d_{2}}}{2}\rfloor$ and
note that $n\leq d_{2}$ and that

$n\geq\lfloor\frac{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}{2}\rfloor\geq^{\min\\{d_{1},d_{3},d_{2}\\}\geq
16}\frac{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}{4}.$

It follows from Theorem 3.12 that for every $\xi\in\operatorname{\mathbb{E}}$
it holds that

$\left\|\rho\left(\underline{X}^{d_{1}^{\prime}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{Y}^{d_{3}^{\prime}}\right)\xi-\rho\left(\underline{X}^{d_{1}^{\prime}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{Y}^{d_{3}^{\prime}}\right)\xi\right\|\lesssim\\\
(d_{2}+m)^{2m}r_{2}^{n}\max_{0\leq d\leq
d_{2}+n-1}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|\leq^{\frac{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}{4}\leq
n\leq d_{2}}\\\
(d_{2}+m)^{2m}r_{2}^{\frac{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}{4}}\max_{0\leq
d\leq
2d_{2}}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|=\\\
(d_{2}+m)^{2m}r^{\sqrt{\min\\{d_{1},d_{3},d_{2}\\}}}\max_{0\leq d\leq
2d_{2}}\max_{\underline{t}\in\mathcal{B}^{d_{2}+1}}\|\xi-\rho(z(2^{d}\underline{t}))\xi\|.$

Next, we will show that the second inequality stated in the theorem follows
from the first one. Define $\phi:\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])\rightarrow\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ to be the isomorphism induced by

$\phi(y(p))=x(p),\phi(x(p))=y(-p),\forall p\in\mathbb{Z}[t_{1},...,t_{m}].$

We note that for every $p\in\mathbb{Z}[t_{1},...,t_{m}]$ it follows that
$\phi(z(p))=z(p)$. Fix $\rho$ and define $\rho_{0}$ to by the action
$\rho\circ\phi$. By the inequality we already proven, it is enough to prove
that for every $\xi\in\operatorname{\mathbb{E}}$ it holds that

$\displaystyle\left\|\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi-\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi\right\|\leq$
$\displaystyle\left\|\rho_{0}\left(\underline{X}^{d_{1}}\right)\rho_{0}\left(\underline{Z}^{d_{2}}\right)\rho_{0}\left(\underline{Y}^{d_{3}}\right)\xi-\rho_{0}\left(\underline{X}^{d_{1}}\right)\rho_{0}\left(\underline{Z}^{d_{2}+1}\right)\rho_{0}\left(\underline{Y}^{d_{3}}\right)\xi\right\|.$

We note that

$\rho_{0}(\underline{Y}^{d_{3}})=\rho(\underline{X}^{d_{3}}),\rho_{0}(\underline{Z}^{d_{2}})=\rho(\underline{Z}^{d_{2}}),\rho_{0}(\underline{Z}^{d_{2}+1})=\rho(\underline{Z}^{d_{2}+1}).$

Also, for every $\underline{t}\in\mathcal{B}^{d_{1}}$

$\displaystyle\rho_{0}(x(-(2^{d_{1}}-1)\underline{t}))\rho_{0}(X^{d_{1}}(\underline{t}))=\rho(y((2^{d_{1}}-1)\underline{t}))\left(\frac{1}{2^{d_{1}}}\sum_{a=0}^{2^{d_{1}}-1}\rho(y(-a\underline{t}))\right)=\rho(Y^{d_{1}}(\underline{t})).$

Thus,

$\left(\prod_{\underline{t}\in\mathcal{B}^{d_{1}}}\rho_{0}(x(-(2^{d_{1}}-1)\underline{t}))\right)\rho_{0}(\underline{X}^{d_{1}})=\rho(\underline{Y}^{d_{1}}).$

It follows that for every $\xi\in\operatorname{\mathbb{E}}$ it holds that

$\left\|\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi-\rho\left(\underline{Y}^{d_{1}}\right)\rho\left(\underline{Z}^{d_{2}+1}\right)\rho\left(\underline{X}^{d_{3}}\right)\xi\right\|=\left\|\left(\prod_{\underline{t}\in\mathcal{B}^{d_{1}}}\rho_{0}(x(-(2^{d_{1}}-1)\underline{t}))\right)\rho_{0}\left(\underline{X}^{d_{1}}\right)\rho_{0}\left(\underline{Z}^{d_{2}}\right)\rho_{0}\left(\underline{Y}^{d_{3}}\right)\xi-\left(\prod_{\underline{t}\in\mathcal{B}^{d_{1}}}\rho_{0}(x(-(2^{d_{1}}-1)\underline{t}))\right)\rho_{0}\left(\underline{X}^{d_{1}}\right)\rho_{0}\left(\underline{Z}^{d_{2}+1}\right)\rho_{0}\left(\underline{Y}^{d_{3}}\right)\xi\right\|\leq^{\text{Claim
}\ref{difference to linear
claim}}\left\|\rho_{0}\left(\underline{X}^{d_{1}}\right)\rho_{0}\left(\underline{Z}^{d_{2}}\right)\rho_{0}\left(\underline{Y}^{d_{3}}\right)\xi-\rho_{0}\left(\underline{X}^{d_{1}}\right)\rho_{0}\left(\underline{Z}^{d_{2}+1}\right)\rho_{0}\left(\underline{Y}^{d_{3}}\right)\xi\right\|,$

as needed. ∎

##  4\. Word norm growth in the $A_{2}$ Steinberg group

Let $\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$ be the Steinberg
group defined above. We note that
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$ is finitely generated
and in particular the set

$S=\bigcup_{1\leq i,j\leq 3,i\neq j}\\{x_{i,j}(\pm 1),x_{i,j}(\pm
t_{1}),...,x_{i,j}(\pm t_{m})\\},$

is a finite generating set.

Below, we will need the following result regarding the growth rate of the word
norm in $\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$:

######  Proposition 4.1.

Let $a\in\mathbb{Z}\setminus\\{0\\},d\in\mathbb{N}\cup\\{0\\}$ be constants
and $\underline{t}\in\mathbb{Z}[t_{1},...,t_{m}]$ be a monomial. Also, let

$S=\bigcup_{1\leq i,j\leq 3,i\neq j}\\{x_{i,j}(\pm 1),x_{i,j}(\pm
t_{1}),...,x_{i,j}(\pm t_{m})\\}.$

For every $1\leq i,j\leq 3,i\neq j$, it holds that

$|x_{i,j}(2^{d}\underline{t})|_{S}\lesssim(1+(d+\deg(\underline{t}))^{2})$

and

$|x_{i,j}(a\underline{t})|_{S}\lesssim(1+\log^{3}(|a|)+(\deg(\underline{t}))^{2})$

where $\log=\log_{2}$.

###### Proof.

We will start by showing that for every $n\in\mathbb{N}$ and every
$p_{1},....,p_{n}\in\mathbb{Z}[t_{1},...,t_{m}]$ it holds that

(4) $\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{n}p_{k}\right)\right|_{S}\leq
4n^{2}\left(\max_{1\leq i,j\leq 3,i\neq j,1\leq k\leq
n}|x_{i,j}(p_{k})|_{S}\right).$

We note that for every $p_{1},p_{2}\in\mathbb{Z}[t_{1},...,t_{m}]$, it holds
that

$|x_{i,j}(p_{1}p_{2})|_{S}=|[x_{i,k}(p_{1}),x_{k,j}(p_{2})]|_{S}\leq
2(|x_{i,k}(p_{1})|_{S}+|x_{k,j}(p_{2})|_{S}).$

Thus,

(5) $\max_{1\leq i,j\leq 3,i\neq j}|x_{i,j}(p_{1}p_{2})|_{S}\leq
2\left(\max_{1\leq i,j\leq 3,i\neq j}|x_{i,j}(p_{1})|_{S}+\max_{1\leq i,j\leq
3,i\neq j}|x_{i,j}(p_{2})|_{S}\right).$

Using (5) it follows that for every $N\in\mathbb{N}$ and every
$p_{1},...,p_{2^{N}}$ it holds that

$\displaystyle\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{2^{N}}p_{k}\right)\right|_{S}\leq$
$\displaystyle 2\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{2^{N-1}}p_{k}\right)\right|_{S}+2\max_{1\leq
i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=2^{N-1}+1}^{2^{N}}p_{k}\right)\right|_{S},$

and by induction on $N$ it follows that

(6) $\displaystyle\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{2^{N}}p_{k}\right)\right|_{S}\leq\sum_{k=1}^{2^{N}}2^{N}\max_{1\leq
i,j\leq 3,i\neq j}\left|x_{i,j}\left(p_{k}\right)\right|_{S}\leq
4^{N}\max_{1\leq i,j\leq 3,i\neq j,1\leq k\leq
n}\left|x_{i,j}\left(p_{k}\right)\right|_{S}.$

We will show by induction on $N$ that for every $1\leq n<2^{N}$ and every
$p_{1},...,p_{n}\in\mathbb{Z}[t_{1},....,t_{n}]$ it holds that

$\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{n}p_{k}\right)\right|_{S}\leq(2^{2N}-2^{N})\left(\max_{1\leq
i,j\leq 3,i\neq j,1\leq k\leq n}|x_{i,j}(p_{k})|_{S}\right).$

If $N=1$, then $n=1$ and $2^{2N}-2^{N}=2^{2}-2^{1}=2$ and the inequality
holds. Next, we will assume that the inequality holds for $N$ and prove it for
$N+1$. Let $1\leq n<2^{N+1}$ and
$p_{1},...,p_{n}\in\mathbb{Z}[t_{1},...,t_{m}]$. If $n<2^{N}$, we are done by
the induction assumption. Also, if $n=2^{N}$, then we are done by (6) and the
fact that $4^{N}\leq 2^{2N+2}-2^{N+1}$.

Thus, we can assume that $2^{N}+1\leq n<2^{N+1}$. Denote $n^{\prime}=n-2^{N}$.
Then $1\leq n^{\prime}<2^{N}$ and by (5) it holds that

$\displaystyle\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{n}p_{k}\right)\right|_{S}\leq$
$\displaystyle 2\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{2^{N}}p_{k}\right)\right|_{S}+2\max_{1\leq
i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=2^{N}+1}^{2^{N}+n^{\prime}}p_{k}\right)\right|_{S}\leq$
$\displaystyle\left(2\cdot 4^{N}+2(2^{2N}-2^{N})\right)\left(\max_{1\leq
i,j\leq 3,i\neq j,1\leq k\leq n}|x_{i,j}(p_{k})|_{S}\right)=$
$\displaystyle(2^{2N+2}-2^{N+1})\left(\max_{1\leq i,j\leq 3,i\neq j,1\leq
k\leq n}|x_{i,j}(p_{k})|_{S}\right),$

as needed.

It follows that for every $N$ and every $2^{N-1}\leq n<2^{N}$ it holds that

$\displaystyle\max_{1\leq i,j\leq 3,i\neq
j}\left|x_{i,j}\left(\prod_{k=1}^{n}p_{k}\right)\right|_{S}\leq
2^{2N}\left(\max_{1\leq i,j\leq 3,i\neq j,1\leq k\leq
n}|x_{i,j}(p_{k})|_{S}\right)\leq 4n^{2}\left(\max_{1\leq i,j\leq 3,i\neq
j,1\leq k\leq n}|x_{i,j}(p_{k})|_{S}\right)$

and thus (4) is proven.

As a consequence (4), we deduce that for
$\underline{t}=2^{d}t_{1}^{n_{1}}...t_{m}^{n_{m}}$, if $d+n_{1}+....+n_{m}>0$,
then

$\displaystyle\max_{1\leq i,j\leq 3,i\neq
j}|x_{i,j}(2^{d}t_{1}^{n_{1}}...t_{m}^{n_{m}})|_{S}\leq
8(d+n_{1}+...+n_{m})^{2}=8(d+\deg(\underline{t}))^{2},$

which proves the first inequality stated in the Proposition.

In order to prove the second inequality stated in the Proposition, it is
enough to show that

$\max_{1\leq i,j\leq 3,i\neq j}|x_{i,j}(a)|_{S}\leq 1+8\log^{3}(|a|),$

for every $a\in\mathbb{Z}\setminus\\{0\\}$. Without loss of generality, we can
assume that $a\in\mathbb{N}$. Fix $a\in\mathbb{N}$ and $1\leq i,j\leq 3,i\neq
j$. We observe that by (4) it holds for every $l\in\mathbb{N}$ that

$|x_{i,j}(2^{l})|_{S}\leq 4l^{2}\left(\max_{1\leq i^{\prime},j^{\prime}\leq
3,i^{\prime}\neq j^{\prime}}|x_{i^{\prime},j^{\prime}}(2)|_{S}\right)\leq
8l^{2}.$

Denote $n=\lfloor\log(a)\rfloor$. Then $2^{n}\leq a<2^{n+1}$ and there are
$a_{0},...,a_{n}\in\\{0,1\\}$ such that $a=\sum_{l}a_{l}2^{l}$. Thus

$\displaystyle|x_{i,j}(a)|_{S}\leq\sum_{l=0}^{n}|x_{i,j}(a_{l}2^{l})|_{S}\leq
a_{0}+\sum_{l=1}^{n}8a_{l}l^{2}\leq 1+8\sum_{l=1}^{n}l^{2}\leq 1+8n^{3}\leq
1+8\log^{3}(a),$

as needed. ∎

######  Corollary 4.2.

Using the notations of section 3, for $i,j\in\\{1,2,3\\},i\neq j$ and
$d\in\mathbb{N}$, we define
$\underline{X}_{i,j}^{d}\in\operatorname{Prob}_{c}(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$
as

$\underline{X}_{i,j}^{d}=\frac{1}{|\operatorname{Poly}(d,d)|}\sum_{p\in\operatorname{Poly}(d,d)}x_{i,j}(p).$

Then for

$S=\bigcup_{1\leq i,j\leq 3,i\neq j}\\{x_{i,j}(\pm 1),x_{i,j}(\pm
t_{1}),...,x_{i,j}(\pm t_{m})\\}$

it holds that

$|\underline{X}_{i,j}^{d}|_{S}\lesssim d^{m+3}.$

###### Proof.

Fix $i,j\in\\{1,2,3\\},i\neq j$. It is enough to show that for every
$p\in\operatorname{Poly}(d,d),p\neq 0$ it holds that

$|x_{i,j}(p)|_{S}\lesssim(d+m)^{m+3}.$

Let
$p=\sum_{\underline{t}\in\mathcal{B}^{d}}c_{\underline{t}}\underline{t}\in\operatorname{Poly}(d,d)\setminus\\{0\\}$.
Then,

$|x_{i,j}(p)|_{S}\leq\sum_{\underline{t}\in\mathcal{B}^{d}}|x_{i,j}(c_{\underline{t}}\underline{t})|_{S}\lesssim^{\text{Proposition
}\ref{unipotent bound
prop}}\sum_{\underline{t}\in\mathcal{B}^{d}}(\log^{3}(c_{\underline{t}})+\deg(\underline{t})^{2})\leq\sum_{\underline{t}\in\mathcal{B}^{d}}(d^{3}+d^{2})\lesssim(d+m)^{m+3}.$

∎

##  5\. Relative fixed point property

In this section, we will prove Theorem 1.2 stated in the introduction. The
heart of the proof is proving relative fixed point property for
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$ and the idea of the
proof in that case is similar to that of the proofs of [Opp22, Theorem 6.1]
and [dLdlS23, Theorem 4.6].

###  5.1. General convergence argument

Let $\operatorname{\mathbb{E}}$ be a Banach space, $D\in\mathbb{N}$ a constant
and
$F_{1},F_{2}:\bigcup_{d\in\mathbb{N}}(\mathbb{N}\cap[d,3d])^{6}\rightarrow\operatorname{\mathbb{E}},\epsilon:(\mathbb{N}\cap[D,\infty))\rightarrow[0,\infty)$
functions such that the following holds:

  1. ($\mathcal{A}$1)

For every $d\in\mathbb{N},d\geq D$ and every
$(d_{1},...,d_{6})\in(\mathbb{N}\cap[d,3d])^{6}$ such that
$d_{2}-(d_{1}+d_{3})\geq d-1$ and $d_{5}-(d_{4}+d_{6})\geq d-1$ it holds that

$\|F_{1}(d_{1},d_{2},d_{3},d_{4},d_{5},d_{6})-F_{2}(d_{3},d_{2},d_{1},d_{6},d_{5},d_{4})\|\lesssim\epsilon(d).$

  2. ($\mathcal{A}$2)

For every $d\in\mathbb{N},d\geq D$, every
$(d_{1},...,d_{6})\in(\mathbb{N}\cap[d,3d])^{6}$ and every $2\leq i\leq 5$, if
$\max\\{d_{i-1},d_{i+1}\\}\geq d_{i}$ and $d_{i}<3d$, then

$\|F_{1}(d_{1},...,d_{i},...,d_{6})-F_{1}(d_{1},...,d_{i}+1,...,d_{6})\|\lesssim\epsilon(d),$

and

$\|F_{2}(d_{1},...,d_{i},...,d_{6})-F_{2}(d_{1},...,d_{i}+1,...,d_{6})\|\lesssim\epsilon(d).$

######  Lemma 5.1.

Let $d\in\mathbb{N},d\geq D$, $(d_{1},...,d_{6})\in(\mathbb{N}\cap[d,3d])^{6}$
and $d^{\prime},d^{\prime\prime}\in\mathbb{N}\cap[d,3d]$.

For $j=1,2$ it holds that

$\|F_{j}(d_{1},d^{\prime},d^{\prime},d_{4},d_{5},d_{6})-F_{j}(d_{1},d^{\prime\prime},d^{\prime\prime},d_{4},d_{5},d_{6})\|\lesssim
d\epsilon(d),$
$\|F_{j}(d_{1},d_{2},d^{\prime},d^{\prime},d_{5},d_{6})-F_{j}(d_{1},d_{2},d^{\prime\prime},d^{\prime\prime},d_{5},d_{6})\|\lesssim
d\epsilon(d),$
$\|F_{j}(d_{1},d_{2},d_{3},d^{\prime},d^{\prime},d_{6})-F_{j}(d_{1},d_{2},d_{3},d^{\prime\prime},d^{\prime\prime},d_{6})\|\lesssim
d\epsilon(d).$

###### Proof.

The proofs of all these inequalities are similar and we will prove only the
first one.

Assume without loss of generality that $d^{\prime}<d^{\prime\prime}$, then it
is enough to prove that

$\|F_{j}(d_{1},d^{\prime},d^{\prime},d_{4},d_{5},d_{6})-F_{j}(d_{1},d^{\prime}+1,d^{\prime}+1,d_{4},d_{5},d_{6})\|\lesssim\epsilon(d).$

Indeed, by $(\mathcal{A}2)$ it holds that

$\|F_{j}(d_{1},d^{\prime},d^{\prime},d_{4},d_{5},d_{6})-F_{j}(d_{1},d^{\prime}+1,d^{\prime}+1,d_{4},d_{5},d_{6})\|\leq\\\
\|F_{j}(d_{1},d^{\prime},d^{\prime},d_{4},d_{5},d_{6})-F_{j}(d_{1},d^{\prime}+1,d^{\prime},d_{4},d_{5},d_{6})\|+\\\
\|F_{j}(d_{1},d^{\prime}+1,d^{\prime},d_{4},d_{5},d_{6})-F_{j}(d_{1},d^{\prime}+1,d^{\prime}+1,d_{4},d_{5},d_{6})\|\lesssim\epsilon(d).$

∎

######  Corollary 5.2.

Let $d\in\mathbb{N},d\geq D$ and
$d_{1},d_{6},d^{\prime},d^{\prime\prime}\in(\mathbb{N}\cap[d,3d])^{6}$.

For $j=1,2$ it holds that

$\|F_{j}(d_{1},d^{\prime},d^{\prime},d^{\prime},d^{\prime},d_{6})-F_{j}(d_{1},d^{\prime\prime},d^{\prime\prime},d^{\prime\prime},d^{\prime\prime},d_{6})\|\lesssim
d\epsilon(d).$

Thus, we can prove the following general convergence result:

######  Theorem 5.3.

Let $\operatorname{\mathbb{E}},D,F_{1},F_{2},\epsilon$ be as above and assume
that $\sum_{d\in\mathbb{N},d\geq D}d\epsilon(d)<\infty$. Define a sequence
$\\{\xi_{k}\\}_{k\in\mathbb{N}}\subseteq\operatorname{\mathbb{E}}$ as follows:

$\xi_{k}=\begin{cases}F_{1}(\frac{k+1}{2},...,\frac{k+1}{2})&k\text{ is
odd}\\\ F_{2}(\frac{k}{2},...,\frac{k}{2})&k\text{ is even}\end{cases}.$

Then this sequence is a Cauchy sequence in $\operatorname{\mathbb{E}}$ and
thus converges.

###### Proof.

By the assumption that $\sum_{d\in\mathbb{N}d\geq D}d\epsilon(d)<\infty$, it
is enough to prove that for $\\{j_{1},j_{2}\\}=\\{1,2\\}$ and $d\geq D$ it
holds that

$\|F_{j_{1}}(d,...,d)-F_{j_{2}}(d+1,...,d+1)\|\lesssim d\epsilon(d).$

Fix $\\{j_{1},j_{2}\\}=\\{1,2\\}$ and $d\geq D$. Then

$\|F_{j_{1}}(d,...,d)-F_{j_{2}}(d+1,...,d+1)\|\lesssim^{\text{Corollary
}\ref{4 in a row coro}}\\\
d\epsilon(d)+\|F_{j_{1}}(d,3d,3d,3d,3d,d)-F_{j_{2}}(d+1,...,d+1)\|\lesssim^{\text{Lemma
}\ref{2 in a row lemma}}\\\
d\epsilon(d)+\|F_{j_{1}}(d,3d,d+1,d+1,3d,d)-F_{j_{2}}(d+1,...,d+1)\|\lesssim^{(\mathcal{A}1)}\\\
d\epsilon(d)+\|F_{j_{2}}(d+1,3d,d,d,3d,d+1)-F_{j_{2}}(d+1,...,d+1)\|\lesssim^{\text{Lemma
}\ref{2 in a row lemma}}\\\
d\epsilon(d)+\|F_{j_{2}}(d+1,3d,3d,3d,3d,d+1)-F_{j_{2}}(d+1,...,d+1)\|\lesssim^{\text{Corollary
}\ref{4 in a row coro}}d\epsilon(d),$

as needed. ∎

###  5.2. Relative fixed point property for
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$

We fix $m\in\mathbb{N}$ and for $1\leq i,j\leq 3,i\neq j$, we denote $K_{i,j}$
to be the $(i,j)$-root subgroup of
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$ as defined above.
Also, for $\\{i,j,k\\}=\\{1,2,3\\}$, we define

$H_{i,k}=\langle
K_{i,j},K_{j,k}\rangle<\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$

and note that $H_{i,k}$ is isomorphic to the group $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$. A key ingredient of the proof of our
relative fixed point property is that we can order the root subgroups such
that each three consecutive root subgroups form a Heisenberg group and the
middle group in the each three consecutive root subgroups is the center of
this Heisenberg group. Explicitly, we define two maps
$\sigma_{1},\sigma_{2}:\\{1,...,6\\}\rightarrow\\{(i_{1},i_{2}):1\leq
i_{1},i_{2}\leq 3,i_{1}\neq i_{2}\\}$ as follows:

$\sigma_{1}(1)=(1,2),\sigma_{1}(2)=(1,3),\sigma_{1}(3)=(2,3),$
$\sigma_{1}(4)=(2,1),\sigma_{1}(5)=(3,1),\sigma_{1}(6)=(3,2),$

and

$\sigma_{2}(1)=(2,3),\sigma_{2}(2)=(1,3),\sigma_{2}(3)=(1,2),$
$\sigma_{2}(4)=(3,2),\sigma_{2}(5)=(3,1),\sigma_{2}(6)=(2,1).$

We note that for every $2\leq i\leq 5$ and every $j=1,2$ it holds that

$\langle
K_{\sigma_{j}(i-1)},K_{\sigma_{j}(i)},K_{\sigma_{j}(i+1)}\rangle=H_{\sigma_{j}(i)},$

and that $K_{\sigma_{j}(i)}$ is the center of $H_{\sigma_{j}(i)}$ (we use a
slight abuse of notation here, i.e., we identify
$K_{i_{1},i_{2}}=K_{(i_{1},i_{2})}$ and $H_{i_{1},i_{2}}=H_{(i_{1},i_{2})}$).

Using the notations of section 3, for $i\in\\{1,...,6\\},j=1,2$ and
$d\in\mathbb{N}$, we define
$\underline{X}_{\sigma_{j}(i)}^{d}\in\operatorname{Prob}_{c}(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$
as

$\underline{X}_{\sigma_{j}(i)}^{d}=\frac{1}{|\operatorname{Poly}(d,d)|}\sum_{p\in\operatorname{Poly}(d,d)}x_{\sigma_{j}(i)}(p).$

With these notations, we can use the results of section 3 and prove the
following:

######  Theorem 5.4.

Let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach space with a
modulus of convexity $\delta:(0,2]\rightarrow(0,1]$ and $r=r(\delta)<1$ be the
constant of Theorem 3.13. Also, let

$S=\bigcup_{1\leq i_{1},i_{2}\leq 3,i_{1}\neq i_{2}}\\{x_{i_{1},i_{2}}(\pm
1),x_{i_{1},i_{2}}(\pm t_{1}),...,x_{i_{1},i_{2}}(\pm t_{m})\\}.$

For every $d\geq\max\\{16,m\\}$, every
$f\in\operatorname{Prob}_{c}(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}]))$,
every affine isometric action $\rho$ of
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$ on
$\operatorname{\mathbb{E}}$ and every unit vector
$\xi\in\operatorname{\mathbb{E}}$ the following holds:

  1. (1)

For $i=2,5$ and $d_{i-1},d_{i},d_{i+1}\in\mathbb{N}\cap[d,3d]$, if
$d_{i}-(d_{i-1}+d_{i+1})\geq d-1$, then

$\left\|\rho(\underline{X}_{\sigma_{1}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{1}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{1}(i+1)}^{d_{i+1}})\rho(f)\xi-\rho(\underline{X}_{\sigma_{2}(i-1)}^{d_{i+1}})\rho(\underline{X}_{\sigma_{2}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{2}(i+1)}^{d_{i-1}})\rho(f)\xi\right\|\lesssim
d^{m}\left(\frac{1}{2}\right)^{d}(d^{2}+|f|_{S}).$

  2. (2)

For $2\leq i\leq 5$, $j=1,2$ and
$d_{i-1},d_{i},d_{i+1}\in\mathbb{N}\cap[d,3d]$, if
$\max\\{d_{i-1},d_{i+1}\\}\geq d_{i}$, then

$\left\|\rho(\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{j}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}})\rho(f)\xi-\rho(\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{j}(i)}^{d_{i}+1})\rho(\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}})\rho(f)\xi\right\|\lesssim
d^{2m}r^{\sqrt{d}}(d^{2}+|f|_{S}).$

###### Proof.

For $2\leq i\leq 5$ and $j=1,2$, we note that $H_{\sigma_{j}(i)}\cong\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ and that the following holds:

  * •

If $i+j$ is odd, then the isomorphism between $H_{\sigma_{j}(i)}$ and $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ is induced by
$x_{\sigma_{j}(i-1)}(p)\mapsto x(p),x_{\sigma_{j}(i+1)}(p)\mapsto
y(p),x_{\sigma_{j}(i)}(p)\mapsto z(p),\forall
p\in\mathbb{Z}[t_{1},...,t_{m}]$. Moreover, extending this isomorphism to the
group ring yields that

$\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}}\mapsto\underline{X}^{d_{i-1}},\underline{X}_{\sigma_{j}(i)}^{d_{i}}\mapsto\underline{Z}^{d_{i}},\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}}\mapsto\underline{Y}^{d_{i+1}}.$

  * •

If $i+j$ is even, then the isomorphism between $H_{\sigma_{j}(i)}$ and $\rm
H_{3}(\mathbb{Z}[t_{1},...,t_{m}])$ is induced by
$x_{\sigma_{j}(i-1)}(p)\mapsto y(p),x_{\sigma_{j}(i+1)}(p)\mapsto
x(p),x_{\sigma_{j}(i)}(p)\mapsto z(p),\forall
p\in\mathbb{Z}[t_{1},...,t_{m}]$. Moreover, extending this isomorphism to the
group ring yields that

$\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}}\mapsto\underline{Y}^{d_{i-1}},\underline{X}_{\sigma_{j}(i)}^{d_{i}}\mapsto\underline{Z}^{d_{i}},\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}}\mapsto\underline{X}^{d_{i+1}}.$

Below, we will use these isomorphisms implicitly in order to apply theorems
from section 3 in our setting.

Fix $d,f,\xi$ as above.

  1. (1)

Assume that $i=2,5$, $d_{i-1},d_{i},d_{i+1}\in\mathbb{N}\cap[d,3d]$ and
$d_{i}-(d_{i-1}+d_{i+1})\geq d-1$. Then

$\left\|\rho(\underline{X}_{\sigma_{1}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{1}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{1}(i+1)}^{d_{i+1}})\rho(f)\xi-\rho(\underline{X}_{\sigma_{2}(i-1)}^{d_{i+1}})\rho(\underline{X}_{\sigma_{2}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{2}(i+1)}^{d_{i-1}})\rho(f)\xi\right\|\lesssim^{\text{Theorem
}\ref{change of order thm},d\geq m}\\\
d^{m}\left(\frac{1}{2}\right)^{d-1}\max_{\underline{t}\in\mathcal{B}^{d_{i}}}\|\rho(f)\xi-\rho(x_{\sigma_{1}(i)}(2^{d_{i}}\underline{t}))\rho(f)\xi\|\lesssim_{S}^{\text{Proposition
}\ref{distance of rho (g) prop},\|\xi\|=1}\\\
d^{m}\left(\frac{1}{2}\right)^{d}\left(\max_{\underline{t}\in\mathcal{B}^{d_{i}}}|x_{x_{\sigma_{1}(i)}}(2^{d_{i}}\underline{t})|_{S}+|f|_{S}\right)\lesssim^{\text{Proposition
}\ref{unipotent bound prop},d_{i}\leq 3d}\\\
d^{m}\left(\frac{1}{2}\right)^{d}\left(d^{2}+|f|_{S}\right),$

as needed.

  2. (2)

Assume that $2\leq i\leq 5$, $d_{i-1},d_{i},d_{i+1}\in\mathbb{N}\cap[d,3d]$
and $\max\\{d_{1},d_{3}\\}\geq d_{2}$. Then

$\left\|\rho(\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{j}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}})\rho(f)\xi-\rho(\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{j}(i)}^{d_{i}+1})\rho(\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}})\rho(f)\xi\right\|\lesssim^{\text{Theorem
}\ref{Final d_1,d_2,d_3 thm},\min\\{d_{i-1},d_{i},d_{i+1}\\}\geq d\geq m}\\\
d_{i}^{2m}r^{\sqrt{d}}\max_{0\leq d^{\prime}\leq
2d_{i}}\max_{\underline{t}\in\mathcal{B}^{d_{i}+1}}\|\rho(f)\xi-\rho(x_{\sigma_{j}(i)}(2^{d^{\prime}}\underline{t}))\rho(f)\xi\|\lesssim^{d_{i}\leq
3d}\\\ d^{2m}r^{\sqrt{d}}\max_{0\leq d^{\prime}\leq
6d}\max_{\underline{t}\in\mathcal{B}^{3d+1}}\|\rho(f)\xi-\rho(x_{\sigma_{j}(i)}(2^{d^{\prime}}\underline{t}))\rho(f)\xi\|\lesssim_{S}^{\text{Proposition
}\ref{distance of rho (g) prop},\|\xi\|=1}\\\
d^{2m}r^{\sqrt{d}}\left(\max_{0\leq d^{\prime}\leq
6d}\max_{\underline{t}\in\mathcal{B}^{3d+1}}|x_{\sigma_{j}(i)}(2^{d^{\prime}}\underline{t})|_{S}+|f|_{S}\right)\lesssim^{\text{Proposition
}\ref{unipotent bound prop},}\\\
d^{2m}r^{\sqrt{d}}\left(d^{2}+|f|_{S}\right),$

as needed.

∎

After this, we will prove the following relative fixed point result:

######  Theorem 5.5.

For every $1\leq i,j\leq 3$, the pair
$(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}],K_{i,j})$ has relative
property $(F_{\mathcal{E}_{uc}})$.

###### Proof.

By symmetry, it is enough to prove that the pair
$(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}],H_{1,3})$ has relative
property $(F_{\mathcal{E}_{uc}})$.

Fix $\operatorname{\mathbb{E}}$ to be a uniformly convex Banach space with a
modulus of convexity $\delta$. We denote $r=r(\delta)<1$ to be the constant of
Theorem 3.13 and note that $r\geq\frac{1}{2}$ (by the choice of
$r_{2}(\delta)$ in Theorem 3.9).

Let $\rho$ to be an affine isometric action of
$\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$ on
$\operatorname{\mathbb{E}}$ and fix a unit vector
$\xi\in\operatorname{\mathbb{E}}$. Also, let

$S=\bigcup_{1\leq i,j\leq 3,i\neq j}\\{x_{i,j}(\pm 1),x_{i,j}(\pm
t_{1}),...,x_{i,j}(\pm t_{m})\\}.$

In order to use Theorem 5.3, we will define
$F_{1},F_{2}:\mathbb{N}^{6}\rightarrow\operatorname{\mathbb{E}}$ as follows:
For the maps $\sigma_{1},\sigma_{2}$ define above and $j=1,2$, we define

$F_{j}(d_{1},...,d_{6})=\rho(\underline{X}_{\sigma_{j}(1)}^{d_{1}})\rho(\underline{X}_{\sigma_{j}(2)}^{d_{2}})\rho(\underline{X}_{\sigma_{j}(3)}^{d_{3}})\rho(\underline{X}_{\sigma_{j}(4)}^{d_{4}})\rho(\underline{X}_{\sigma_{j}(5)}^{d_{5}})\rho(\underline{X}_{\sigma_{j}(6)}^{d_{6}})\xi.$

We further define a function $\epsilon:\mathbb{N}\rightarrow[0,\infty)$ as
$\epsilon(d)=d^{3m+3}r^{\sqrt{d}}$. We will show that for $D=\max\\{16,m\\}$,
the functions $F_{1},F_{2}$ fulfil the conditions $(\mathcal{A}1)$ and
$(\mathcal{A}2)$ defined in section 5.1.

First, we will verify that $(\mathcal{A}1)$ holds. Fix $d\geq D$ and
$(d_{1},...,d_{6})\in(\mathbb{N}\cap[d,3d])^{6}$ such that
$d_{2}-(d_{1}+d_{3})\geq d-1$ and $d_{5}-(d_{4}+d_{6})\geq d-1$. Define
$F_{3}:\mathbb{N}^{6}\rightarrow\operatorname{\mathbb{E}}$ as

$F_{3}(d_{1},...,d_{6})=\rho(\underline{X}_{\sigma_{2}(1)}^{d_{1}})\rho(\underline{X}_{\sigma_{2}(2)}^{d_{2}})\rho(\underline{X}_{\sigma_{2}(3)}^{d_{3}})\rho(\underline{X}_{\sigma_{1}(4)}^{d_{4}})\rho(\underline{X}_{\sigma_{1}(5)}^{d_{5}})\rho(\underline{X}_{\sigma_{1}(6)}^{d_{6}})\xi.$

We note that

$\|F_{1}(d_{1},d_{2},d_{3},d_{4},d_{5},d_{6})-F_{2}(d_{3},d_{2},d_{1},d_{6},d_{5},d_{4})\|\leq\\\
\|F_{1}(d_{1},d_{2},d_{3},d_{4},d_{5},d_{6})-F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})\|+\\\
\|F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})-F_{2}(d_{3},d_{2},d_{1},d_{6},d_{5},d_{4})\|.$

Thus, it is enough to prove that

$\|F_{1}(d_{1},d_{2},d_{3},d_{4},d_{5},d_{6})-F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})\|\lesssim\epsilon(d)$

and

$\|F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})-F_{2}(d_{3},d_{2},d_{1},d_{6},d_{5},d_{4})\|\lesssim\epsilon(d).$

For the first inequality, we denote
$f=\underline{X}_{\sigma_{1}(4)}^{d_{4}}\underline{X}_{\sigma_{1}(5)}^{d_{5}}\underline{X}_{\sigma_{1}(6)}^{d_{6}}\in\operatorname{Prob}_{c}(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$
and note that

$|f|_{S}\lesssim^{\text{Corollary }\ref{vert X_i,j vert_S
coro},d_{4},d_{5},d_{6}\leq 3d}d^{m+3}.$

Then

$\|F_{1}(d_{1},d_{2},d_{3},d_{4},d_{5},d_{6})-F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})\|=\\\
\left\|\rho(\underline{X}_{\sigma_{1}(1)}^{d_{1}})\rho(\underline{X}_{\sigma_{1}(2)}^{d_{2}})\rho(\underline{X}_{\sigma_{1}(3)}^{d_{3}})\rho(f)\xi-\rho(\underline{X}_{\sigma_{2}(1)}^{d_{3}})\rho(\underline{X}_{\sigma_{2}(2)}^{d_{2}})\rho(\underline{X}_{\sigma_{1}(3)}^{d_{1}})\rho(f)\xi\right\|\lesssim^{\text{Theorem
}\ref{bounding norm moves thm}}\\\
d^{m}\left(\frac{1}{2}\right)^{d}\left(d^{2}+|f|_{S}\right)\lesssim\left(\frac{1}{2}\right)^{d}(d^{m+2}+d^{2m+3})\lesssim\epsilon(d).$

The proof of the bound on
$\|F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})-F_{2}(d_{3},d_{2},d_{1},d_{6},d_{5},d_{4})\|$
is somewhat simpler:

$\|F_{3}(d_{3},d_{2},d_{1},d_{4},d_{5},d_{6})-F_{2}(d_{3},d_{2},d_{1},d_{6},d_{5},d_{4})\|\leq^{\text{Claim
}\ref{difference to linear claim}}\\\
\left\|\rho(\underline{X}_{\sigma_{1}(4)}^{d_{4}})\rho(\underline{X}_{\sigma_{1}(5)}^{d_{5}})\rho(\underline{X}_{\sigma_{1}(6)}^{d_{6}})\xi-\rho(\underline{X}_{\sigma_{2}(4)}^{d_{6}})\rho(\underline{X}_{\sigma_{1}(5)}^{d_{5}})\rho(\underline{X}_{\sigma_{1}(6)}^{d_{4}})\xi\right\|\lesssim^{\text{Theorem
}\ref{bounding norm moves thm}}\\\
d^{m}\left(\frac{1}{2}\right)^{d}d^{2}\leq\epsilon(d).$

Thus, we proved that $(\mathcal{A}1)$ holds.

Next, we will prove that $(\mathcal{A}2)$ holds. Fix $j\in\\{1,2\\}$,
$d\in\mathbb{N},d\geq D$, $(d_{1},...,d_{6})\in(\mathbb{N}\cap[d,3d])^{6}$ and
$2\leq i\leq 5$. Assume that $\max\\{d_{i-1},d_{i+1}\\}\geq d_{i}$.

We denote
$f\in\operatorname{Prob}_{c}(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$
to be

$f=\underline{X}_{\sigma_{j}(i+2)}^{d_{i+2}}...\underline{X}_{\sigma_{j}(6)}^{d_{6}}$

(in the case where $i=5$, $f=\delta_{e}$). We note that

$|f|_{S}\lesssim^{\text{Corollary }\ref{vert X_i,j vert_S coro}}d^{m+3}.$

Then,

$\|F_{j}(d_{1},...,d_{i},...,d_{6})-F_{j}(d_{1},...,d_{i}+1,...,d_{6})\|\leq^{\text{Claim
}\ref{difference to linear claim}}\\\
\left\|\rho(\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{j}(i)}^{d_{i}})\rho(\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}})\rho(f)\xi-\rho(\underline{X}_{\sigma_{j}(i-1)}^{d_{i-1}})\rho(\underline{X}_{\sigma_{j}(i)}^{d_{i}+1})\rho(\underline{X}_{\sigma_{j}(i+1)}^{d_{i+1}})\rho(f)\xi\right\|\lesssim^{\text{Theorem
}\ref{bounding norm moves thm}}\\\
d^{2m}r^{\sqrt{d}}\left(d^{2}+|f|_{S}\right)\lesssim
d^{2m}r^{\sqrt{d}}(d^{2}+d^{m+3})\lesssim d^{3m+3}r^{\sqrt{d}}=\epsilon(d).$

Thus, we proved that $(\mathcal{A}2)$ is fulfilled.

As in Theorem 5.3, we will define

$\xi_{k}=\begin{cases}F_{1}(\frac{k+1}{2},...,\frac{k+1}{2})&k\text{ is
odd}\\\ F_{2}(\frac{k}{2},...,\frac{k}{2})&k\text{ is even}\end{cases}.$

By Theorem 5.3 it follows that this is a Cauchy sequence and thus converges.
Denote $\widetilde{\xi}=\lim\xi_{k}$. We will show that
$\widetilde{\xi}\in\operatorname{\mathbb{E}}^{\rho(H_{1,3})}$ and thus
$\operatorname{\mathbb{E}}^{\rho(H_{1,3})}\neq\emptyset$.

Fix $\underline{t}_{0}$ to be a monomial. For $d\geq\deg(\underline{t}_{0})$
and $j=1,2$, we denote
$f_{j}^{d}\in\operatorname{Prob}_{c}(\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$
to be the probability function such that

$\underline{X}_{\sigma_{j}(1)}^{d}\underline{X}_{\sigma_{j}(2)}^{d}\underline{X}_{\sigma_{j}(3)}^{d}\underline{X}_{\sigma_{j}(4)}^{d}\underline{X}_{\sigma_{j}(5)}^{d}\underline{X}_{\sigma_{j}(6)}^{d}=\left(\frac{1}{2^{d}}\sum_{a=0}^{2^{d}-1}x_{\sigma_{j}(1)}(a\underline{t}_{0})\right)f_{j}^{d}.$

Thus,

${\left\|\rho(x_{\sigma_{j}(1)}(\underline{t}_{0}))F_{j}(d,....,d)-F_{j}(d,....,d)\right\|=}\\\
\left\|\rho(x_{\sigma_{j}(1)}(\underline{t}_{0}))\rho\left(\frac{1}{2^{d}}\sum_{a=0}^{2^{d}-1}x_{\sigma_{j}(1)}(a\underline{t}_{0})\right)\rho(f_{j}^{d})\xi-\rho\left(\frac{1}{2^{d}}\sum_{a=0}^{2^{d}-1}x_{\sigma_{j}(1)}(a\underline{t}_{0})\right)\rho(f_{j}^{d})\xi\right\|\lesssim^{\text{Proposition
}\ref{ergodic diff prop}}\\\
\frac{1}{2^{d}}\left\|\rho(x_{\sigma_{j}(1)}(2^{d}\underline{t}_{0}))\rho(f_{j}^{d})\xi-\rho(f_{j}^{d})\xi\right\|\lesssim_{S}^{\text{Proposition
}\ref{distance of rho (g) prop},\|\xi\|=1}\\\
\frac{1}{2^{d}}\left(|x_{\sigma_{j}(1)}(2^{d}\underline{t}_{0})|_{S}+|f_{j}^{d}|_{S}\right)\lesssim^{\text{Proposition
}\ref{unipotent bound prop},\text{Corollary }\ref{vert X_i,j vert_S
coro}}\frac{d^{m+3}}{2^{d}}.$

It follows that

$\|\rho(x_{\sigma_{1}(1)}(\underline{t}_{0}))\widetilde{\xi}-\widetilde{\xi}\|=\lim_{d\rightarrow\infty}\|\rho(x_{\sigma_{1}(1)}(\underline{t}_{0}))\xi_{2d-1}-\xi_{2d-1}\|=\\\
\lim_{d\rightarrow\infty}\|\rho(x_{\sigma_{1}(1)}(\underline{t}_{0}))F_{1}(d,....,d)-F_{1}(d,....,d)\|\lesssim\lim_{d\rightarrow\infty}\frac{d^{m+3}}{2^{d}}=0,$

and thus
$\rho(x_{\sigma_{1}(1)}(\underline{t}_{0}))\widetilde{\xi}=\widetilde{\xi}$.
This holds for every $\underline{t}_{0}$ and thus
$\widetilde{\xi}\in\operatorname{\mathbb{E}}^{\rho(K_{1,2})}$. Similarly,

$\|\rho(x_{\sigma_{2}(1)}(\underline{t}_{0}))\widetilde{\xi}-\widetilde{\xi}\|=\lim_{d\rightarrow\infty}\|\rho(x_{\sigma_{2}(1)}(\underline{t}_{0})\xi_{2d}-\xi_{2d}\|=\\\
\lim_{d\rightarrow\infty}\|\rho(x_{\sigma_{2}(1)}(\underline{t}_{0}))F_{2}(d,....,d)-F_{2}(d,....,d)\|=0,$

and thus
$\rho(x_{\sigma_{2}(1)}(\underline{t}_{0}))\widetilde{\xi}=\widetilde{\xi}$.
This holds for every $\underline{t}_{0}$ and thus
$\widetilde{\xi}\in\operatorname{\mathbb{E}}^{\rho(K_{2,3})}$.

We conclude that

$\widetilde{\xi}\in\operatorname{\mathbb{E}}^{\rho(K_{1,2})}\cap\operatorname{\mathbb{E}}^{\rho(K_{2,3})}=\operatorname{\mathbb{E}}^{\rho(H_{1,3})}$

as needed. ∎

###  5.3. Relative fixed point property for $\operatorname{St}_{\Phi}$

Below, for a subgroup $K<\Gamma$ and $g\in\Gamma$, we denote $K^{g}=gKg^{-1}$.
Also, given a root $\gamma\in\Phi$ and a ring $R$, we will denote

$K_{\gamma}(R)=\\{x_{\gamma}(p):p\in R\\}.$

######  Observation 5.6.

We note that if the pair $(\Gamma,K)$ has relative property
$(F_{\mathcal{E}_{uc}})$ and $\phi:\Gamma\rightarrow\Gamma$ is an isomorphism
of $\Gamma$, then the $(\Gamma,\phi(K))$ has relative property
$(F_{\mathcal{E}_{uc}})$. In particular, if the pair $(\Gamma,K)$ has relative
property $(F_{\mathcal{E}_{uc}})$, then for every $g\in\Gamma$, the pair
$(\Gamma,K^{g})$ has relative property $(F_{\mathcal{E}_{uc}})$.

######  Theorem 5.7.

If for some root $\gamma_{0}\in C_{2}$, the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma_{0}})$ has
relative property $(F_{\mathcal{E}_{uc}})$, then for every $\gamma\in C_{2}$,
the pair $(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma})$
has relative property $(F_{\mathcal{E}_{uc}})$.

###### Proof.

We note that for every $\gamma,\gamma^{\prime}\in C_{2}$ such that $\gamma$
and $\gamma^{\prime}$ are both short/long roots, there is an automorphism
$\phi:\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}])\rightarrow\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$
such that $\phi(K_{\gamma})=K_{\gamma^{\prime}}$. This implies that if
$\gamma,\gamma^{\prime}$ are both both short/long roots, then, by Observation
5.6, the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$ if and only if the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma^{\prime}})$
has relative property $(F_{\mathcal{E}_{uc}})$.

Let $\gamma_{0}\in C_{2}$ such that the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma_{0}})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

#### Case 1: $\gamma_{0}$ is a long root

By our argument above, it follows that for every long root $\gamma$, the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}],K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$. We are left to show that there is
a short root $\gamma$ such that the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

Explicitly, let $\alpha$ denote a long root of $C_{2}$ and $\beta$ denote a
short root of $C_{2}$ such that the angle between $\alpha$ and $\beta$ is
$\frac{3}{4}\pi$. We already showed that
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha})$ and
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta})$
have relative property $(F_{\mathcal{E}_{uc}})$. We will show that the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+\beta})$
has relative property $(F_{\mathcal{E}_{uc}})$.

By Proposition 2.14, it holds for every $p\in\mathbb{Z}[t_{1},...,t_{m}]$ that

$[x_{\alpha}(p),x_{\beta}(1)]=x_{\alpha+\beta}(p)x_{\alpha+2\beta}(p).$

Thus,

$x_{\alpha+\beta}(p)=[x_{\alpha}(p),x_{\beta}(1)]x_{\alpha+2\beta}(-p)=x_{\alpha}(-p)(x_{\beta}(-1)x_{\alpha}(p)x_{\beta}(1))x_{\alpha+2\beta}(-p)\subseteq
K_{\alpha}K_{\alpha}^{x_{\beta}(-1)}K_{\alpha+2\beta},$

and it follows that

$K_{\alpha+\beta}\subseteq
K_{\alpha}K_{\alpha}^{x_{\beta}(-1)}K_{\alpha+2\beta}$

and in particular, $K_{\alpha+\beta}$ is boundedly generated by $K_{\alpha}$,
$K_{\alpha}^{x_{\beta}(-1)}$ and $K_{\alpha+2\beta}$.

Thus, $K_{\alpha+\beta}$ is boundedly generated by $3$ subgroups that all have
relative property $(F_{\mathcal{E}_{uc}})$ and it follows from Lemma 2.6 that
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+\beta})$
has relative property $(F_{\mathcal{E}_{uc}})$.

#### Case 2: $\gamma_{0}$ is a short root

The proof of this case is similar to the proof of the previous case (but it is
more involved). By our argument above, it follows that for every short root
$\gamma$, the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$. We are left to show that there is
a long root $\gamma$ such that the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

Explicitly, for $\alpha,\beta$ as above, we need to show that the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta})$
has relative property $(F_{\mathcal{E}_{uc}})$.

A monomial $\underline{t}=t_{1}^{k_{1}}...t_{m}^{k_{m}}$ will be called
square-free if $k_{i}\in\\{0,1\\}$ for every $1\leq i\leq m$. We note that for
every monomial $\underline{t}=t_{1}^{k_{1}}...t_{m}^{k_{m}}$, there is a
unique square-free monomial, which we will denote by
$\underline{t}_{\text{sf}}$ and a unique monomial $\underline{t}_{\text{rt}}$
such that
$\underline{t}=\underline{t}_{\text{sf}}\underline{t}_{\text{rt}}^{2}$. We
note that there are exactly $2^{m}$ different square-free monomials. For a
square-free monomial $\underline{t}^{\prime}$, we will denote
$(\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}$ to be polynomials
that are of the form $a_{1}\underline{t}_{1}+...+a_{n}\underline{t}_{n}$ such
that for every $1\leq i\leq n$,
$(\underline{t}_{i})_{\text{sf}}=\underline{t}^{\prime}$. We define

$K_{\alpha+2\beta}((\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}})=\\{x_{\alpha+2\beta}(p):p\in(\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}\\},$

and note that $K_{\alpha+2\beta}$ is boundedly generated by
$K_{\alpha+2\beta}((\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}),\underline{t}^{\prime}\text{
square-free}$. Thus, it is enough to show that for every square-free
$\underline{t}^{\prime}$, it holds that the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta}((\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}))$
has relative property $(F_{\mathcal{E}_{uc}})$.

Fix $\underline{t}^{\prime}$ to be a square-free monomial. By Proposition
2.14, it holds for every $p_{1}\in\mathbb{Z}[t_{1},...,t_{m}]$ that

$[x_{\alpha+\beta}(p_{1}),x_{\beta}(1)]=x_{\alpha+2\beta}(2p_{1})$

and thus $K_{\alpha+2\beta}(2\mathbb{Z}[t_{1},...,t_{m}])$ is boundedly
generated by $K_{\alpha+\beta}$ and $K_{\beta}$.

Also, by Proposition 2.14, it holds for every
$p_{2}\in\mathbb{Z}[t_{1},...,t_{m}]$ that

$[x_{\alpha}(\underline{t}^{\prime}),x_{\beta}(p_{2})]=x_{\alpha+\beta}(\underline{t}^{\prime}p_{2})x_{\alpha+2\beta}(\underline{t}^{\prime}p_{2}^{2}).$

Thus, the set

$\\{x_{\alpha+2\beta}(\underline{t}^{\prime}p_{2}^{2}):p_{2}\in\mathbb{Z}[t_{1},...,t_{m}]\\}$

is boundedly generated by $K_{\beta},K_{\alpha+\beta}$ and
$K_{\beta}^{x_{\alpha}(-\underline{t}^{\prime})}$.

It follows that it is enough to show that for every
$p\in(\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}$, there are
$p_{1},p_{2}\in\mathbb{Z}[t_{1},...,t_{m}]$ such that
$p=2p_{1}+\underline{t}^{\prime}p_{2}^{2}$. Indeed, if such $p_{1},p_{2}$
exist for every $p\in(\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}$,
then it follows that
$K_{\alpha+2\beta}((\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}})$ is
boundedly generated by $K_{\beta},K_{\alpha+\beta}$ and
$K_{\beta}^{x_{\alpha}(-\underline{t}^{\prime})}$ and by Lemma 2.6, the pair
$(\operatorname{St}_{C_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta}((\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}))$
has relative property $(F_{\mathcal{E}_{uc}})$.

Fix $p\in(\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}$. We note that
there are monomials
$\underline{t}_{1},...,\underline{t}_{n}\in(\mathbb{Z}[t_{1},...,t_{m}])_{\underline{t}^{\prime}}$
such that
$p+2\mathbb{Z}[t_{1},...,t_{m}]=\sum_{i=1}^{n}\underline{t}_{i}+2\mathbb{Z}[t_{1},...,t_{m}]$.
As noted above, for every $i$, there is a monomial
$(\underline{t}_{i})_{\text{rt}}$ such that
$\underline{t}_{i}=\underline{t}^{\prime}(\underline{t}_{i})_{\text{rt}}^{2}$.
Thus,

$p+2\mathbb{Z}[t_{1},...,t_{m}]=\sum_{i=1}^{n}\underline{t}_{i}+2\mathbb{Z}[t_{1},...,t_{m}]=\underline{t}^{\prime}\sum_{i=1}^{n}(\underline{t}_{i})_{\text{rt}}^{2}+2\mathbb{Z}[t_{1},...,t_{m}]=\underline{t}^{\prime}\left(\left(\sum_{i=1}^{n}(\underline{t}_{i})_{\text{rt}}\right)^{2}-2\sum_{1\leq
i<j\leq
n}(\underline{t}_{i})_{\text{rt}}(\underline{t}_{j})_{\text{rt}}\right)+2\mathbb{Z}[t_{1},...,t_{m}]=\underline{t}^{\prime}\left(\sum_{i=1}^{n}(\underline{t}_{i})_{\text{rt}}\right)^{2}+2\mathbb{Z}[t_{1},...,t_{m}].$

It follows that there are $p_{1},p_{2}\in\mathbb{Z}[t_{1},...,t_{m}]$ such
that $p=2p_{1}+\underline{t}^{\prime}p_{2}^{2}$ as needed. ∎

The proof for relative fixed point property in the case where $\Phi=G_{2}$
appeared in [EJZK17] in the context of property (T). The adaptation of the
proof of [EJZK17] to our setting is straight-forward and we claim no
originality here.

######  Theorem 5.8.

For every root $\gamma\in G_{2}$, the pair
$(\operatorname{St}_{G_{2}}(\mathbb{Z}[t_{1},...,t_{m}],K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

###### Proof.

Using the notations of Proposition 2.14, we note that
$K_{\\{\pm\alpha,\pm(\alpha+3\beta),\pm(2\alpha+3\beta)\\}}\cong\operatorname{St}_{A_{2}}(\mathbb{Z}[t_{1},...,t_{m}])$.
By Theorem 5.5, we deduce that for every long root $\gamma\in G_{2}$, it holds
that pair
$(\operatorname{St}_{G_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\gamma})$ has
relative property $(F_{\mathcal{E}_{uc}})$. Thus, we are left to prove
relative property $(F_{\mathcal{E}_{uc}})$ for subgroups of short roots in
$G_{2}$. By symmetry, it is enough to show that the pair
$(\operatorname{St}_{G_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta})$
has relative property $(F_{\mathcal{E}_{uc}})$.

By Proposition 2.14, for every $p_{1},p_{2}\in\mathbb{Z}[t_{1},...,t_{m}]$ it
holds that

$[x_{\alpha}(p_{1}),x_{\beta}(p_{2})]=x_{\alpha+\beta}(p_{1}p_{2})x_{\alpha+2\beta}(p_{1}p_{2}^{2})x_{\alpha+3\beta}(p_{1}p_{2}^{3})x_{2\alpha+3\beta}(p_{1}^{2}p_{2}^{3}),$

and

$[x_{\alpha}(p_{1}p_{2}),x_{\beta}(1)]=x_{\alpha+\beta}(p_{1}p_{2})x_{\alpha+2\beta}(p_{1}p_{2})x_{\alpha+3\beta}(p_{1}p_{2})x_{2\alpha+3\beta}(p_{1}^{2}p_{2}^{2}).$

Thus,

$x_{\alpha+2\beta}(p_{1}(p_{2}^{2}-p_{2}))\in[x_{\alpha}(p_{1}p_{2}),x_{\beta}(1)]^{-1}[x_{\alpha}(p_{1}),x_{\beta}(p_{2})]K_{\alpha+3\beta}K_{2\alpha+3\beta}\subseteq
K_{\alpha}^{x_{\beta}(1)}K_{\alpha}K_{\alpha}^{x_{\beta}(-p_{2})}K_{\alpha+3\beta}K_{2\alpha+3\beta}.$

Taking $p_{1}=p\in\mathbb{Z}[t_{1},...,t_{m}]$ to be a general polynomial and
$p_{2}=2,t_{1},t_{2},...,t_{m}$, we deduce that

$K_{\alpha+2\beta}(2\mathbb{Z}[t_{1},...,t_{m}])\subseteq
K_{\alpha}^{x_{\beta}(1)}K_{\alpha}K_{\alpha}^{x_{\beta}(-2)}K_{\alpha+3\beta}K_{2\alpha+3\beta},$
$K_{\alpha+2\beta}((t_{i}^{2}-t_{i})\mathbb{Z}[t_{1},...,t_{m}])\subseteq
K_{\alpha}^{x_{\beta}(1)}K_{\alpha}K_{\alpha}^{x_{\beta}(-t_{i})}K_{\alpha+3\beta}K_{2\alpha+3\beta},\forall
1\leq i\leq m.$

Denote

$K_{\alpha+2\beta}^{\prime}=K_{\alpha+2\beta}\left(2\mathbb{Z}[t_{1},...,t_{m}]+\sum_{i=1}^{m}(t_{i}^{2}-t_{i})\mathbb{Z}[t_{1},...,t_{m}]\right).$

By the computations above, $K_{\alpha+2\beta}^{\prime}$ is boundedly generated
by
$K_{\alpha},K_{\alpha+3\beta},K_{2\alpha+3\beta},K_{\alpha}^{x_{\beta}(1)},K_{\alpha}^{x_{\beta}(-2)}$
and $K_{\alpha}^{x_{\beta}(-t_{i})},i=1,...,m$. It follows from Lemma 2.6 that
$(\operatorname{St}_{G_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta}^{\prime})$
has relative property $(F_{\mathcal{E}_{uc}})$. We note that
$K_{\alpha+2\beta}^{\prime}$ is a finite index subgroup of $K_{\alpha+2\beta}$
(explicitly,
$K_{\alpha+2\beta}/K_{\alpha+2\beta}^{\prime}=\oplus_{i=1}^{2^{m}}(\mathbb{Z}/2\mathbb{Z})$)
and thus it follows that
$(\operatorname{St}_{G_{2}}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha+2\beta})$
has relative property $(F_{\mathcal{E}_{uc}})$ as needed. ∎

Combining these results and Theorem 5.5 give rise to the following result that
appeared in the introduction:

######  Theorem 5.9.

Let $\Phi\neq C_{2}$ be a classical, reduced, irreducible root system of rank
$\geq 2$ and $R$ a finitely generated commutative (unital) ring. For every
$\alpha\in\Phi$, the pair $(\operatorname{St}_{\Phi}(R),K_{\alpha})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

###### Proof.

Fix $\Phi$ and $R$ as above.

We note that there is an integer $m\in\mathbb{N}$ such that there is a ring
epimorphism $\phi:\mathbb{Z}[t_{1},...,t_{m}]\rightarrow R$. This ring
epimorphism induces a group epimorphism
$\widetilde{\phi}:\operatorname{St}_{\Phi}(\mathbb{Z}[t_{1},...,t_{m}])\rightarrow\operatorname{St}_{\Phi}(R)$
such that every $\alpha\in\Phi$,
$\widetilde{\phi}(K_{\alpha}(\mathbb{Z}[t_{1},...,t_{m}]))=K_{\alpha}(R)$.
Thus, it is enough to prove that for every $\alpha\in\Phi$, the pair
$(\operatorname{St}_{\Phi}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

#### Case 1: $\Phi$ is simply laced, $\Phi=F_{4}$ or $\Phi=G_{2}$

In this case, every root $\alpha$ of $\Phi$ is either in an $A_{2}$ subsystem
or in a $G_{2}$ system (if $\Phi=G_{2}$) and thus by Theorems 5.5, 5.8, for
every $\alpha\in\Phi$, the pair
$(\operatorname{St}_{\Phi}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha})$ has
relative property $(F_{\mathcal{E}_{uc}})$.

#### Case 2: $\Phi=B_{n},n\geq 3$ or $\Phi=C_{n},n\geq 3$

In this case, the only roots that are not in an $A_{2}$ subsystem are the
short roots in the $C_{2}$ subsystem (if $\Phi=B_{n}$) or the long roots in
the $C_{2}$ subsystem (if $\Phi=C_{n}$). By Theorem 5.7, it follows that for
every $\alpha$ the pair
$(\operatorname{St}_{\Phi}(\mathbb{Z}[t_{1},...,t_{m}]),K_{\alpha})$ has
relative property $(F_{\mathcal{E}_{uc}})$. ∎

##  6\. General synthesis machinery

The aim of this section is to develop a general machinery that allows one to
deduce property $(F_{\mathcal{E}_{uc}})$ for a group $\Gamma$ from relative
property $(F_{\mathcal{E}_{uc}})$ given a certain subgroup structure of
$\Gamma$. This machinery was inspired by the work of Mimura [Mim16] (which in
turn was inspired by the work of Shalom [Sha06]), but our machinery is much
more general than that of [Mim16].

###  6.1. Ultraproduct of Banach spaces and group actions

We recall that an ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is a non-empty
collections of subsets of $\mathbb{N}$ such that the following holds:

  * •

$\emptyset\notin\mathcal{U}$.

  * •

If $B\in\mathcal{U}$ and $B\subseteq A$, then $A\in\mathcal{U}$.

  * •

If $A,B\in\mathcal{U}$, then $A\cap B\in\mathcal{U}$.

  * •