# On the critical regularity of nilpotent groups acting on the interval: the
metabelian case

Maximiliano Escayola and Cristóbal Rivas

###### Abstract

Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian group.
In this note we show that $G$ embeds into the group of orientation preserving
$C^{1+\alpha}$-diffeomorphisms of the closed interval, for all $\alpha<1/k$
where $k$ is the torsion-free rank of $G/A$ and $A$ is a maximal abelian
subgroup. We show that in many situations, this embedding has critical
regularity in the sense that there is no embedding of $G$ with higher
regularity. A particularly nice family where this critical regularity is
achieved, is the family of $(2n+1)$-dimensional Heisenberg groups, where we
can show that its critical regularity equals $1+1/n$.

Keywords: Nilpotent groups, Heisenberg groups, diffeomorphism groups, Hölder
continuity, critical regularity.

2020 Mathematics Subject Classification: 20F18, 37C05, 37C85, 37E05.

##  1 Introduction

Given an integer $n\geq 0$ and a real number $\alpha\in[0,1)$, we denote by
$\text{Diff}_{+}^{\,n+\alpha}([0,1])$ the group of all orientation preserving
$C^{n}$-diffeomorphisms of the closed interval $[0,1]$ whose $n$-th derivative
is $\alpha$-Holder continuous ($C^{1+\alpha}$-diffemorphisms for short).
Remark that with this notation the group $\text{Diff}_{+}^{\,0}([0,1])$ is the
group of all homemomorfisms of $[0,1]$ isotopic to the identity. Finally,
observe that the family of groups $\text{Diff}_{+}^{\,n+\alpha}([0,1])$ is
totally ordered by inclusion since
$\text{Diff}_{+}^{\,n+\alpha}([0,1])\supseteq\text{Diff}_{+}^{\,n^{\prime}+\alpha^{\prime}}([0,1])$
if and only if $n<n^{\prime}$, or $n=n^{\prime}$ and
$\alpha\leq\alpha^{\prime}$.

We are interested on computing the critical regularity of an abstract group
$G$ acting on the interval $[0,1]$. Recall that, given a group $G$, the
critical regularity of $G$ on $[0,1]$ is by definition

$\text{Crit}_{[0,1]}(G)=\text{sup}\\{n+\alpha\mid n\geq 0,\alpha\in[0,1)\text{
and }G\text{ embeds into }\text{Diff}_{+}^{\,n+\alpha}([0,1])\\},$

where we set $\text{Crit}_{[0,1]}(G)=-\infty$ if $G$ does not embeds into
$\text{Diff}_{+}^{0}([0,1])$. The problem of computing the critical regularity
of a group $G$ is quite natural and turns out to be very interesting in the
case that $G$ is finitely-generated (the reader may wish to consult [16] for
an introduction). For instance, we know from a theorem of Deroin, Klepstyn,
Navas [8] (see also [9]) that every countable subgroup of
$\text{Diff}^{0}_{+}([0,1])$ is conjugated to a group of bilipchitz
transformations, and hence $1\leq\text{Crit}_{[0,1]}(G)$ for every countable
subgroup of $\text{Diff}^{0}_{+}([0,1])$ (for uncountable subgroups
$\text{Diff}^{0}_{+}([0,1])$ this is no longer true, see [5]). However, the
celebrated Stability Theorem of Thurston [27] implies that every finitely-
generated group of $\text{Diff}^{1}_{+}([0,1])$ admits a surjective
homomorphisms onto the integers, and so not every group of homeomorphisms of
the interval can be realized as a group of diffeomorphisms111Concrete examples
of finitely-generated subgroups of $\text{Diff}_{+}^{\,0}([0,1])$ having
trivial abelianization can be found in [27, 2, 25]. However, Thuston’s
obstruction is not the only obstruction for $C^{1}$ smoothability as there are
also known examples of finitely-generated and locally indicable groups having
no faithful $C^{1}$ action on the interval, see [5, 22, 3, 17].. Further
obstructions appears in higher regularity: for $C^{2}$ there is the important
Kopell’s obstruction [18], and between $C^{1}$ and $C^{2}$ there are the
generalized Kopell’s obstruction from [8]. In a related spirit, Kim and
Koberda [15], and later Mann and Wolff [19], have shown that for every $n\geq
1$ and every $\alpha$ in $[0,1)$, there is a finitely-generated group whose
critical regularity on $[0,1]$ is exactly $n+\alpha$.

In this work, we focus on actions on the interval of finitely-generated and
torsion-free nilpotent groups. Let $G$ be one such group. It follows from the
work of Mal’cev that $G$ embeds into $\text{Diff}^{0}_{+}([0,1])$ (see, for
instance, [7, §5.2] and [10, §1.2]), and we know from the work of Farb and
Franks [11], that every action of $G$ on $[0,1]$ by homeomorphisms can be
conjugated inside $\text{Diff}^{1}_{+}([0,1])$ (see also the universal
contruction from Jorquera [13]). This was further refined by Parkhe [23] who
showed that actually $G$ can be conjugated inside
$\text{Diff}^{1+\alpha}_{+}([0,1])$ as long as $\alpha<1/\tau$, where $\tau$
is the degree of the polinomial growth of the nilpotent group $G$. On the
other hand, Plante and Thurston [26], have shown that every nilpotent subgroup
of $\text{Diff}^{2}_{+}([0,1])$ must be abelian. So, if $G$ is a torsion-free,
finitely-generated and nilpotent group which is non-abelian, then

$1+1/\tau\leq\text{Crit}_{[0,1]}(G)\leq 2.$

The exact critical regularity of concrete nilpotent groups has been computed
only in few cases and one important goal of this work is to provide new
explicit computations of critical regularity for certain groups. Castro,
Jorquera and Navas [6], build a family of nilpotent abelian-by-cyclic groups
whose critical regularity is 2. These examples can be made of arbitrarily
large nilpotency degree, yet they are all metabelian (i.e. its commutator
subgroup is abelian). Jorquera, Navas and the second author showed in [14]
that the critical regularity of $N_{4}$ (the group of $4$ by $4$ upper
triangular matrices with $1$’s in the diagonal) is $1+1/2$. We point out that
at the time of this writing, $N_{4}$ is the only torsion-free nilpotent group
whose critical regularity is computed and turns out not to be an integer.
Remark that $N_{4}$ is also a metabelian group.

The main purpose of this article is to exhibit many other nilpotent groups
whose critical regularity is strictly between 1 and 2. Our main technical
result is an improvement of Parkhe’s lower bound for the critical regularity
in the class of torsion-free nilpotent groups which are metabelian (see Remark
1.1). For the statement recall that the torsion-free rank of an abelian group
$A$ is the dimension of the $\mathbb{Q}$-vector space $A\otimes\mathbb{Q}$. We
denote this rank by $\text{rank}(A)$.

######  Theorem A.

Let $G$ be a torsion-free, finitely-generated nilpotent group which is
metabelian, and let $A$ be a maximal abelian subgroup containing $[G,G]$. If
$k=\text{rank}(G/A)$, then

$G\hskip 5.69046pt\text{embeds into}\hskip
5.69046pt\textup{Diff}_{+}^{1+\alpha}([0,1]),\hskip 5.69046pt\text{for
all}\hskip 5.69046pt\alpha<1/k.$

In particular $1+1/k\leq\text{Crit}_{[0,1]}(G)$.

######  Remark 1.1.

By Bass-Guivarc’h formula [1, 12], the degree of the polynomial growth of a
nilpotent group $G$ is $\tau=\sum_{i\geq 1}i\text{
rank}(\gamma_{i}/\gamma_{i+1})$, where
$G=\gamma_{1}\geqslant\gamma_{2}\geqslant...$ is the lower central series of
$G$. In particular, for a nilpotent group $G$ as in Theorem A with maximal
abelian subgroup $A$, we have that $\text{rank}(G/A)<\tau$. Hence the lower
bound for $\text{Crit}_{[0,1]}(G)$ in Theorem A is (strictly) greater than
Parkhe’s lower bound (in the non-abelian case).

The proof of Theorem A is given in Section 3. In §3.1 we build, for a
metabelian and torsion-free nilpotent group $G$, a family of actions of $G$ on
the interval $[0,1]$ by orientation preserving homeomorphisms. This is done by
first building actions of $G$ on $\mathbb{Z}^{k}$ which preserve a
lexicographic order of $\mathbb{Z}^{k}$ and then projecting them into the
interval. In §3.2, we use the Pixton-Tsuboi technique [24, 28] to show that
these actions can be smoothed to actions by $C^{1+\alpha}$-diffeomorphisms for
any $\alpha<1/k$. This subsection closely follows the works [6] and [14] and
the main difference is that we don’t have explicit polinomials but only bounds
on them (see Proposition 3.1). Although these actions may not be faithful, in
§3.3 we explain how to glue some of these actions in order to obtain an
embedding of $G$ into $\textup{Diff}_{+}^{1+\alpha}([0,1])$ for any
$\alpha<1/k$.

In some situations even the lower bound in Theorem A is not sharp in the sense
that there are groups for which the theorem applies yet its critical
regularity is strictly grater than the corresponding $1+1/k$. This is related
to the possibility of splitting the group as a product of two groups each
allowing an embedding with higher regularity. We provide an easy example of
this phenomenon in §4.3. However, in many cases we can ensure that the
inequality in Theorem A is indeed optimal, and in §4.1 and §4.2 we provide two
families of examples were we can obtain upper bounds for the regularity and
hence compute its critical regularity.

The first family of examples are the $(2n+1)$-dimensional discrete Heisenberg
groups, that we denote $\mathscr{H}_{n}$. Recall that by definition

$\mathscr{H}_{n}:=\left\\{\begin{pmatrix}1&\vec{x}&c\\\
\vec{0}^{\;t}&I_{n}&\vec{y}^{\;t}\\\
0&\vec{0}&1\end{pmatrix}\;:\vec{x},\vec{y}\in\mathbb{Z}^{n}\text{ and
}c\in\mathbb{Z}\right\\},$

where $I_{n}$ denotes the $(2n-1)$ identity matrix and $\vec{0}^{\;t}$,
$\vec{y}^{\;t}$ are the transposes of $\vec{0}$, $\vec{y}$ respectively. It is
easy to see that these groups are nilpotent of degree two and hence they are
metabelian. Moreover, a maximal abelian subgroup $A$ of $\mathscr{H}_{n}$ is
given by the set of matrices whose corresponding vector $\vec{x}=0$. In
particular $\mathscr{H}_{n}/A$ has torsion-free rank equal to $n$. For this
family we show in §4.1 that there is no embedding of $\mathscr{H}_{n}$ into
$\text{Diff}_{+}^{1+\alpha}([0,1])$ for $\alpha>1/n$. In particular we obtain

######  Theorem B.

Let $\mathscr{H}_{n}$ be the $(2n+1)$-dimensional discrete Heisenberg group.
Then

$\textup{Crit}_{[0,1]}(\mathscr{H}_{n})=1+\frac{1}{n}.$

Finally, in §4.2 we produce examples of metabelian and torsion-free nilpotent
groups where we can compute its critical regularity but its nilpotency degree
can be chosen to be arbitrarily large. More precisely we show

######  Theorem C.

For any integers $k$ and $d$ with $d>k$, there is a nilpotent group $G$ and a
maximal abelian subgroup $A$ containing $[G,G]$ such that $d$ is the
nilpotency degree of $G$, $k$ is de torsion-free rank of $G/A$ and

$\textup{Crit}_{[0,1]}(G)=1+\frac{1}{k}.$

In both cases, the key to obtain an upper bound for the regularity is to use
the internal algebraic structure of the groups in order to be able to apply to
generalized Kopell lemma from [8].

######  Remark 1.2.

We know from the results of Kim and Koberda [15] that for any real number
$\alpha\geq 1$ there is a finitely-generated group whose critical regularity
is exactly $\alpha$. However, in all known cases where the critical regularity
of a torsion-free nilpotent group has being computed, it is of the form
$1+1/n$ for some integer $n$. See Theorems B, C and [6], [14]. So, we wonder
if this is always the case for torsion-free nilpotent groups (non necesarily
metabelian).

##### Acknowledgements:

The first author acknowledges the hospitality of the Institut de Mathématique
de Bourgone where part of this project was carried out. In particular he is
grateful to Michele Triestino for fruitful discussion around this and other
topics of mathematics. He is also greatful to Nicolás Matte Bon for
discussions around the algebraic structure and actions of nilpotent groups.

The first author acknowledges the partial support of Project Gromeov
ANR-19-CE40-0007 and Mathamsud 210020 ”Dynamical Groups Theory”. Both authors
are grateful to Andrés Navas form reading a prelimiray version of this work.
Both authors acknowledge the support of FONDECYT 1210155.

##  2 Preliminars on nilpotent metabelian groups

Given a group $G$ and two elements $f,g\in G$, we let $[f,g]=fgf^{-1}g^{-1}$
denotes the commutator of $f$ and $g$. Further, if $G$ is finitely-generated
and $S$ is a finite generating set, an element of the form $[s_{1},s_{2}]$
with $s_{1},s_{2}\in S$ is called a simple commutator of weight 2.
Inductively, a _simple commutator of weight_ $n$ is defined as an element of
the form

$[s_{1},...,s_{n}]:=[s_{1},[s_{2},...,s_{n}]],\hskip
11.38092pts_{1},...,s_{n}\in S.$

Note that given $n$, there are only a finite number of simple commutators of
weight $n$.

Let $H$ and $K$ be subgroups of $G$. $[K,H]$ denotes the subgroup of $G$
generated by all commutators $[g,h]$ with $g\in K$ and $h\in H$. The subgroup
$[G,G]$ is called the commutator subgroup and we say that $G$ is metabelian if
$[G,G]$ is abelian.

Remember that the _lower central series_ of $G$ is

$G=\gamma_{0}\geqslant\gamma_{1}\geqslant\gamma_{2}\geqslant\cdots$

where $\gamma_{1}=[G,G]$ and $\gamma_{i}=[G,\gamma_{i-1}]$; and the _upper
central series_ of $G$ is

$\\{e\\}=\zeta_{0}\leqslant\zeta_{1}\leqslant\zeta_{2}\leqslant\cdots$

where $\zeta_{i}/\zeta_{i-1}=Z\left(G/\zeta_{i-1}\right)$, and $Z(G)$ denotes
the center of $G$.

Let us recall some classic results about Nilpotent groups. See [7] for an in-
deep exposition of them. The group $G$ is _nilpotent_ of degree $n$ if
$\zeta_{n}=G$ but $\zeta_{n-1}\neq G$. In this case it also happens that
$\gamma_{n}=\\{e\\}$ but $\gamma_{n-1}\neq\\{e\\}$. Therefore, in a finitely-
generated nilpotent group of degree $n$, we only have a finite number of
simple commutators (for a fixed generator), this is because all simple
commutators of weight $n$ are trivial.

It is a result of Mal’cev that if $G$ is a torsion-free nilpotent group then
the factors $\zeta_{i}/\zeta_{i-1}$ are also torsion-free for all
$i\in\\{1,...,n\\}$ (see [7, Proposition 5.2.19]). Note also that every
subgroup of a finitely-generated nilpotent group is finitely-generated as
well. This is clear for abelian groups and this fact is preserved under group
extension, so it holds for nilpotent groups. Additionally nilpotent groups
also satisfy that their non-trivial normal subgroups always intersect non-
trivially the center of the group (see [7, Proposition 5.2.1]). An immediate
consequence of this is the following useful result.

######  Proposition 2.1.

Let $G$ be a nilpotent group and $\varphi:G\rightarrow H$ be a group
homomorphism. Then $\varphi$ is inyective if and only if $\varphi\mid_{Z(G)}$
(the restriction of $\varphi$ to $Z(G)$) is inyective.

For $g\in G$, Centr$(g)=\\{h\in G\mid gh=hg\\}$ denotes the centralizer of
$G$. The following proposition, although elementary, will be very important to
build actions of $G$ in $\mathbb{Z}^{k}$ in §3.1.

######  Proposition 2.2.

Let $G$ be a torsion-free and finitely-generated nilpotent metabelian group.
Then:

  * •

Given $g\in G$ and $0\neq m\in\mathbb{Z}$ we have that
$\textup{Centr}(g^{m})=\textup{Centr}(g).$

  * •

Let $A\leqslant G$ be a maximal abelian subgroup. If $A$ is normal in $G$,
then $G/A$ is torsion-free.

Proof. Assume that there exist $f,g\in G$, $m\in\mathbb{Z}$ such that
$[f,g^{m}]=e\text{ but }[f,g]\neq e$, and define $H:=\langle f,g\rangle$, the
subgroup generated by $f$ and $g$. Since $[g,g^{k}]=[f,g^{k}]=e$ we have that
$g^{k}\in Z(H)$, and since $H/Z(H)$ is torsion-free then $g\in Z(H)$, which is
a contradiction because $f$ and $g$ do not commute. Then
$\text{Centr}(g^{m})\subseteq\text{Centr}(g)$ (the other contention is
obvious).

The second point follows from the first. Let $A$ be a maximal abelian subgroup
which is also normal, and assume $G/A$ is not torsion-free. Say $g\in G$ is
such that $g\notin A$ but $g^{m}\in A$ for some $m\neq 0$. Then, since $A$ is
abelian, we have that $A\subseteq\text{Centr}(g^{m})=$Centr$(g)$. In
particular $\langle A,g\rangle$, the group generated by $A$ and $g$, is an
abelian subgroup larger than $A$, contradicting our assumption.

$\hfill\square$

###  2.1 On the action of $G/A$ on $A$

Let $G$ be a torsion-free nilpotent group of degree $n$ which is also
metabelian. Let $A$ be a maximal abelian subgroup which we assume that
contains $[G,G]$ (in particular it is normal). In view of Proposition 2.2, we
have that $G$ is an extension of $\mathbb{Z}^{k}$ by $\mathbb{Z}^{d}$,

$1\longrightarrow\mathbb{Z}^{d}\longrightarrow
G\longrightarrow\mathbb{Z}^{k}\longrightarrow 1,$

where $A\simeq\mathbb{Z}^{d}$ and $G/A\simeq\mathbb{Z}^{k}$. In this section
we study the the natural action of $G/A$ on $A$ coming from the conjugacy
action of $G$ on $A$.

Let $\\{g_{1},...,g_{d}\\}$ and $\\{f_{1}A,...,f_{k}A\\}$ be generators of $A$
and $G/A$ respectively. Since $A$ is normal, the subgroup of $G$ generated by
$f_{1},\ldots,f_{k}$, acts on $A$ by automorphisms

$\langle f_{1},...,f_{k}\rangle\longrightarrow\text{Aut}(\mathbb{Z}^{d}).$

Therefore, the action of each $f\in\langle f_{1},...,f_{k}\rangle$ is given by
a matrix ${A}_{f}\in GL_{d}(\mathbb{Z})$, which depends on the generator
$\\{g_{1},...,g_{d}\\}$. We call $A_{f}$ the _conjugacy matrix_ of $f$. In the
special case of the generators $f_{1},\ldots,f_{k}$, we denote the conjugacy
matrix of $f_{i}$ simply by $A_{i}$.

In the next lemma, we will see that we can always choose a generator of $A$,
such that the conjugacy matrices of the elements $f_{1},\ldots,f_{k}$ belong
to $U_{d}(\mathbb{Z})$, the group of upper triangular matrices with 1’s in the
diagonal. This is know to Mal’cev in the case the matrix coefficient belongs
to a field. We write a direct proof in our special case. For the statement we
will say that a generating set of a group is minimal, if it has least posible
cardinality.

######  Lemma 2.3.

Let $A\leqslant G$ be a maximal abelian subgroup satisfying that
$[G,G]\subseteq A$. Suppose that $\mathbb{Z}^{d}\simeq A$ and
$\mathbb{Z}^{k}\simeq G/A=\langle f_{1}A,...,f_{k}A\rangle$. Then, there
exists a generating set $\\{g_{1},...,g_{d}\\}$ of $A$, such that the
conjugacy matrices of the elements $f_{1},\ldots,f_{k}$ belong to
$U_{d}(\mathbb{Z})$. In particular, the nilpotency degree of $G$ is bounded by
$d+1$.

Proof. Since $G$ is nilpotent of degree $n$, the upper central series

$\\{e\\}=\zeta_{0}\leqslant\zeta_{1}\leqslant\cdots\leqslant\zeta_{n}=G,$

is finite. Remember that all the factors $\zeta_{i}/\zeta_{i-1}$ are torsion-
free. Combining this with the fact that $G/A$ is also torsion-free, we have ,
for $g\in G$, that

$g^{j}\in\zeta_{i}\cap A\Rightarrow g\in\zeta_{i}\cap A\hskip 8.5359pt\forall
i\in\\{0,...,n\\},j\in\mathbb{Z}.$ (1)

Define $\Gamma_{i}:=\zeta_{i}\cap A$ and let $m$ be the smallest element in
$\\{1,...,n\\}$ such that $\Gamma_{m}=A$. This yields the filtration

$\\{e\\}=\Gamma_{0}\leqslant\Gamma_{1}\leqslant\cdots\leqslant\Gamma_{m}=A,$

such that

$[G,\Gamma_{i}]\subseteq\Gamma_{i-1},$ (2)

and, by (1), also enjoys the property that each factor
$\Gamma_{i+1}/\Gamma_{i}$ is torsion-free abelian.

Note that if $\Gamma_{m-1}\simeq\mathbb{Z}^{n_{m-1}}$ and
$\Gamma_{m}/\Gamma_{m-1}\simeq\mathbb{Z}^{n_{m}}$ then, since
$\Gamma_{m}=A\simeq\mathbb{Z}^{d}$ is abelian, we have that $d=n_{m-1}+n_{m}$.
Therefore, if $\\{g_{1},\ldots,g_{n_{m-1}}\\}$ and
$\\{g_{n_{m-1}+1}\Gamma_{m-1},\ldots,g_{n_{m-1}+n_{m}}\Gamma_{m-1}\\}$ are
minimal generating sets of $\Gamma_{m-1}$ and $\Gamma_{m}/\Gamma_{m-1}$
respectively, then $\\{g_{1},\ldots,g_{n_{m-1}},g_{n_{m-1}+1},\ldots,g_{d}\\}$
is a minimal generating set of $\Gamma_{m}=A$.

Recursively, we obtain a minimal generating set $\\{g_{1},\ldots,g_{d}\\}$ of
$A$ which, by (2), has the property that for
$g_{s}\in\\{g_{1},...,g_{d}\\}\cap\Gamma_{i}$, it holds that
$[f_{j},g_{s}]\in\Gamma_{i-1}\subseteq\langle g_{1},...,g_{s-1}\rangle$ for
all $j\in\\{1,...,k\\}.$ In other words the conjugacy matrices of each
$f\in\\{f_{1},\ldots,f_{k}\\}$ belongs to $U_{d}(\mathbb{Z})$.

The fact that $G$ has nilpotency degree bounded by $d+1$ follows from the fact
that $U_{d}(\mathbb{Z})$ has nilpotency degree $d+1$.$\hfill\square$

###  2.2 Invariant orders and actions by homeomorphisms

Before we continue, let us take note of some tools related to orderable
groups. A group $G$ is left-orderable if it admits a total order relation, say
$\preceq$, which is invariant under multiplication from the left, that is

$\text{if}\hskip 5.69046ptf\preceq g\hskip 5.69046pt\text{then}\hskip
5.69046pthf\preceq hg\hskip 5.69046pt\text{for all}\hskip 5.69046pth\in G.$

If additionally, $\preceq$ is also invariant under multiplication from the
right, we say that $\preceq$ is a bi-invariant order (bi-order for short).

What is important for this work is that a countable group is left-orderable if
and only if it embeds into $\text{Diff}_{+}^{0}([0,1])$ (see [20, §2] or [10,
§1.1.3] for details). More generally, given a group $G$ that acts on a
countable and totally order set $(\Omega,\preceq)$ by order preserving
bijections, say $\omega\mapsto g(\omega)$, for $g\in G$ and $\omega\in\Omega$,
then there is a dynamical realization of this action. This means that there is
an order preserving map $i:(\Omega,\preceq)\to([0,1],\leq)$ and an
homomorphism $\psi:G\to\text{Diff}_{+}^{0}([0,1])$ satisfying that

$\psi(g)(i(\omega))=i(g(\omega))$

for every $\omega\in\Omega$ and every $g\in G$. See [4, Lemma 2.40] for a
proof. Clearly $\psi$ is an embedding whenever the $G$ action on $\Omega$ is
faithful.

Given a left-order $\preceq$ on a group $G$, we say that a subset $S\subseteq
G$ is _convex_ if for all $h\in G$ and $f,g\in S$ satisfying that $f\preceq
h\preceq g$, we have that $h\in S$. In other words, $S$ is an interval. The
notion of convexity is useful to build orders on group and to build set
carrying invariant orders. This is the content of the following two lemmata.

######  Lemma 2.4 (Convex extension).

If $G$ contains a normal subgroup $A$ such that both $A$ and $G/A$ are left-
orderable. Then $G$ admits a left-order $\preceq$ for which $A$ is convex.

Proof. Indeed, letting $\preceq_{0}$ and $\preceq_{1}$ be left-orders on $A$
and $G/A$, respectively, we may define $\preceq$ on $G$ by

$f\prec g\text{ if either }fA\prec_{1}gA\text{ or }fA=gA\text{ and
}e\prec_{0}f^{-1}g.$

$\hfill\square$

######  Lemma 2.5 (Induced order on cosets).

Let $(G,\preceq)$ be a left-ordered group, and let $H\leqslant G$ be a convex
subgroup. Then the set $G/H$ carries a natural total order which is invariant
under left-multiplication by $G$.

Proof. Indeed, $\leq is$ given by

$fH<gH\hskip 8.5359pt\text{if}\hskip 8.5359ptfh_{1}\prec gh_{2}\hskip
14.22636pt\text{for some}\hskip 14.22636pth_{1},h_{2}\in H.$

Checking that $\leq$ is a well defined and $G$-invariant total order is
routine and is left to the reader.

$\hfill\square$

An important family of examples of left-orderable groups are finitely-
generated and torsion-free abelian groups. Indeed we will repetitively use the
lexicographic order of $\mathbb{Z}^{n}$: we say that

$(i_{1},\ldots,i_{n})\prec(i_{1}^{\prime},\ldots,i_{n}^{\prime})\hskip
5.69046pt\Leftrightarrow\hskip 5.69046pt\exists\>k\in\\{1,\ldots,n\\}\hskip
5.69046pt\text{ such that }\hskip 5.69046pti_{k}<i_{k}^{\prime}\hskip
5.69046pt\text{ and }\hskip 5.69046pti_{s}=i_{s}^{\prime}\hskip
5.69046pt\text{ for }\hskip 5.69046pts<k.$ (3)

######  Remark 2.6.

From this point is easy to see that finitely-generated and torsion-free
nilpotent groups are left-orderable. Indeed, using the upper central series
(which has torsion-free abelian successive quotient) and applying Lemma 2.4
successively, one obtains a total left-order on the nilpotent group.

Further, it is well known that a group is left-orderable if and only if all
its finitely-generated subgroups are left-orderable, see [10, §1.1.2], hence
any torsion-free nilpontent group admits a left-order.

##  3 Proof of Theorem A

Throughout this hole section, $G$ will denote a torsion-free nilpotent group
which is also metabelian and $A$ a maximal abelian subgroup which we assume
that contains $[G,G]$ (in particular it is normal). Recall that then $G$ is an
extension of $G/A\simeq\mathbb{Z}^{k}$ by $A\simeq\mathbb{Z}^{d}$. In
particular the nilpotency degree of $G$ is bounded by $d+1$, see Lemma 2.3.

###  3.1 An action of $G$ on a totally ordered set

######  Proposition 3.1.

Let $A\leqslant G$ be a maximal abelian subgroup satisfying $[G,G]\subseteq
A$. Fix a generating set $\\{g_{1},...,g_{d},f_{1},...,f_{k}\\}$ of $G$, such
that, $\\{g_{1},...,g_{d}\\}$ is the generating set of $A$ given by Lemma 2.3
and $\langle f_{1}A,...,f_{k}A\rangle=G/A\simeq\mathbb{Z}^{k}$. Then, for a
fixed $s\in\\{1,...,d\\}$ there is an action of $G$ on $\mathbb{Z}^{k+1}$
satisfying:

  1. 1)

The action of $G$ on $\mathbb{Z}^{k+1}$ preserve the lexicographic order.

  2. 2)

There exist functions $\ell_{t},r_{m}:\mathbb{Z}^{k}\rightarrow\mathbb{Z}$,
such that

$f_{t}(i_{1},..,i_{t},..,i_{k},j)=(i_{1},..,i_{t}+1,..,i_{k},j+\ell_{t}(i_{1},...,i_{k})),$
$g_{m}(i_{1},...,i_{k},j)=(i_{1},...i_{k},j+r_{m}(i_{1},...,i_{k})),$

for all $m\in\\{1,...,d\\}$ and all $t\in\\{1,...,k\\}$. Besides, $r_{s}\equiv
1$ and $r_{1}=r_{2}=\cdots=r_{s-1}\equiv 0$.

  3. 3)

There exist a positive constant $M$, such that for all $t\in\\{1,...,k\\}$,
$m\in\\{1,...,d\\}$ and

$(i_{1},...,i_{k})\neq(0,...,0)$ we have

$\lvert\ell_{t}(i_{1},...,i_{k})\rvert\leq M(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d},\hskip 14.22636pt\lvert r_{m}(i_{1},...,i_{k})\rvert\leq
M(\lvert i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d}.$

Proof. We first show item 2. To this end, fix $s\in\\{1,...,d\\}$ and consider
the subgroup
$H_{s}=\langle\\{g_{1},...,g_{d}\\}\smallsetminus\\{g_{s}\\}\rangle$. Since
the sets $\\{f_{1}^{i_{1}}\cdots
f_{k}^{i_{k}}A:\;i_{1},...,i_{k}\in\mathbb{Z}\\}$ and
$\\{g_{s}^{j}H_{s}:j\in\mathbb{Z}\\}$ are partitions of $G$ and $A$
respectively, the coset space can be described by the normal forms

$G/H_{s}=\\{f_{1}^{i_{1}}\cdots
f_{k}^{i_{k}}g_{s}^{j}H_{s}\;:\;i_{1},...,i_{k},j\in\mathbb{Z}\\}.$ (4)

Hence we can identify $G/H_{s}$ with $\mathbb{Z}^{k+1}$ (as sets) by
identifying $f_{1}^{i_{1}}\cdots f_{k}^{i_{k}}g_{s}^{j}H_{s}$ with
$({i_{1}},\cdots,{i_{k}},j)$ . In particular, the left-multiplication action
of $G$ on $G/H_{s}$ provides an action of $G$ on $\mathbb{Z}^{k+1}$. This is
the action we want to consider.

Now, by Lemma 2.3, we have that for all $i,j\in\\{1,...,k\\}$ and
$l\in\\{1,...,d\\}$ it holds that

$f_{i}f_{j}\in f_{j}f_{i}\langle g_{1},...,g_{d}\rangle\hskip
5.69046pt\text{and}\hskip 5.69046ptg_{l}f_{j}\in f_{j}g_{l}\langle
g_{1},...,g_{l-1}\rangle.$

Therefore, for $t\in\\{1.\ldots,k\\}$, the action of $f_{t}$ is addition by 1
on the $t$ coordinate and the and the action on the $k+1$ coordinate depends
on previous $k$ coordinates, hence the function $\ell_{t}$. The function
$r_{m}$, for $m\in\\{1,\ldots,d\\}$, can be found analogously.

We are now in position to check item 1. This follows from the fact that the
lexicographic order on the coset space $\mathbb{Z}^{k+1}\simeq G/H_{s}$ given
by (3), can recovered as a convex extension of the lexicographic order on
$\mathbb{Z}^{k}\simeq G/A$ by the (lexicographic) order on
$A/H_{s}\simeq\mathbb{Z}$. See Lemma 2.4 and Lemma 2.5.

Finally we check item 3. Let $t\in\\{1,...,k\\}$. Recall that the action of
$f_{t}$ on $\mathbb{Z}^{k+1}$ is nothing but the left-multiplication action of
$f_{t}$ on $G/H_{s}$. Hence, in order to compute the image of
$f_{1}^{i_{1}}\cdots f_{t}^{i_{t}}\cdots f_{k}^{i_{k}}g_{s}^{j}H_{s}$ under
$f_{t}$, we need to multiply and find the representative in normal form (4).
To do this, observe that $f_{t}f_{j}=[f_{t},f_{j}]f_{j}f_{t}$. Hence, bringing
$f_{t}$ to the $t$-th position, generate at most $\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert$ simple commutators of weight $2$, which
we now need to move to the right most place. Since $G$ is metabelian, the
commutators commute with each other. So, moving them all to the rightmost
place generates at most $(\lvert i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{2}$
simple commutators of weight $3$. Analogously, moving them all to the
rightmost place, we have at most $(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{3}$ simple commutators of weight 4, and so on. Since $G$ has
nilpotency degree bounded by $d+1$, all simple commutators of this weight are
trivial (see Lemma 2.3). Therefore, repeating the previous argument $d+1$
times, we have

$f_{t}.(f^{i_{1}}_{1}\cdots f_{t}^{i_{t}}\cdots
f_{k}^{i_{k}}g_{s}^{j}H_{s})=f^{i_{1}}_{1}\cdots f_{t}^{i_{t}+1}\cdots
f_{k}^{i_{k}}gg_{s}^{j}H_{s},$

where $g\in A$ is the product of at most

$\sum_{i=1}^{d}(\lvert i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{i}\leq d(\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d}$

simple commutators. Now note that

$g.g_{s}^{j}H_{s}=g_{s}^{\ell_{t}(i_{1},...,i_{k})}g_{s}^{j}H_{s},$

since $\ell_{t}(i_{1},...,i_{k})$ agrees with the exponent of $g_{s}$ in the
expression of $g$ over the generators $\\{g_{1},...,g_{d}\\}$. Therefore,
letting $\mathcal{S}\subseteq A$ be the set of all simple commutators of $G$
(which is finite), and defining

$\lambda:=\max\\{\lvert m_{s}\rvert:\exists\;m_{1},...,m_{d}\text{ for which
}(g_{1}^{m_{1}}\cdots g_{s}^{m_{s}}\cdots g_{d}^{m_{d}})\in\mathcal{S}\\},$

we see that $\ell_{t}(i_{1},\ldots,i_{k})$ is bounded by $\lambda$ times the
number of simple commutators that where use to write $g$. Hence

$\lvert\ell_{t}(i_{1},...,i_{k})\rvert\leq\lambda d(\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d}.$

Analogous computations give the inequality for the functions $r_{m}$.

$\hfill\square$

######  Remark 3.2.

Note that the action built in the Proposition 3.1 is not necessarily faithful.
However, given $g_{s}\in\\{g_{1},...,g_{d}\\}$ we have that the action
$G\curvearrowright\mathbb{Z}^{k+1}$ is such that the elements
$g_{1},...,g_{s-1}$ act trivially and
$g_{s}(i_{1},...,i_{k},j)=(i_{1},...,i_{k},j+1)$. This will be used in Section
3.3 in order to build a faithful action.

###  3.2 Action by diffeomorphisms of $[0,1]$

In the previous section we built and action of $G$ on $\mathbb{Z}^{k+1}$ which
preserves the lexicographic order, and hence we can consider the dynamical
realization of this action to get a $G$-action by orientation preserving
homeomorphisms of $[0,1]$.

However, since the group is nilpotent and we have good control from the
polynomials appearing in Proposition 3.1, we will see that this action can
actually be projected to an action by diffeomorphisms of $[0,1]$. For this we
need the following result from Pixton and Tsuboi [24, 28].

######  Lemma 3.3.

There exists a family of $C^{\infty}$ diffeomorphisms
$\varphi_{I^{\prime},I}^{J^{\prime},J}:I\rightarrow J$ between intervals $I$,
$J$, where $I^{\prime}$ (resp. $J^{\prime}$) is an interval contiguous to $I$
(resp. $J$) by the left, such that:

  1. 1)

For all $I^{\prime},I$, $J^{\prime},J,$ $K^{\prime},K$ as above,

$\varphi_{J^{\prime},J}^{K^{\prime},K}\circ\varphi_{I^{\prime},I}^{J^{\prime},J}=\varphi_{I^{\prime},I}^{K^{\prime},K}.$

  2. 2)

For all $I^{\prime},I$, $J^{\prime},J$,

$D\varphi^{J^{\prime},J}_{I^{\prime},I}(x_{-})=\frac{\lvert
J^{\prime}\rvert}{\lvert I^{\prime}\rvert}\text{ and
}D\varphi^{J^{\prime},J}_{I^{\prime},I}(x_{+})=\frac{\lvert J\rvert}{\lvert
I\rvert},$

where $x_{-}$ (resp. $x_{+}$) is the left (resp. right) endpoint of $I$.

  3. 3)

There is a constant $M$ such that for all $x\in I$, we have

$D\textup{log}(D\varphi_{I^{\prime},I}^{J^{\prime},J})(x)\leq\frac{M}{\lvert
I\rvert}\left\lvert\frac{\lvert I\rvert\lvert J^{\prime}\rvert}{\lvert
J\rvert\lvert I^{\prime}\rvert}-1\right\rvert.$

  4. 4)

Given $I^{\prime},I,\;J^{\prime},J,\;K^{\prime},K,\>L^{\prime},L$, as above,
then

$\left\lvert\textup{log}(D\varphi_{I^{\prime},I}^{K^{\prime},K})(x)-\textup{log}(D\varphi_{J^{\prime},J}^{L^{\prime},L})(y)\right\rvert\leq\left\lvert\textup{log}\frac{\lvert
K\rvert\lvert J\rvert}{\lvert I\rvert\lvert
L\rvert}\right\rvert+\left\lvert\textup{log}\frac{\lvert
K^{\prime}\rvert\lvert I\rvert}{\lvert I^{\prime}\rvert\lvert
K\rvert}\right\rvert+\left\lvert\textup{log}\frac{\lvert
L^{\prime}\rvert\lvert J\rvert}{\lvert J^{\prime}\rvert\lvert
L\rvert}\right\rvert,$

for all $x\in I$, $y\in J$.

Now, let
$\\{I_{i_{1},...,i_{k},j}:(i_{1},\ldots,i_{k},j)\in\mathbb{Z}^{k+1}\\}$ be a
family of intervals whose union is dense in $[0,1]$ and that are disposed
preserving the lexicographic order of $\mathbb{Z}^{k+1}$. We identify the
generators $g_{1},\ldots,g_{d},f_{1},\ldots,f_{k}$ from Lemma 2.3 with
elements in $\text{Diff}^{0}_{+}([0,1])$ as follows: $f_{t}$ and $g_{s}$ will
be homeomorphisms of $[0,1]$, such that their restriction to
$I_{i_{1},...,i_{k},j}$ coincides, respectively, with

$\varphi_{I_{i_{1},...,i_{k},j-1}\,,\,I_{i_{1},...,i_{k},j}}^{I_{i_{1},..,i_{t}+1,..,i_{k},j+\ell_{t}(i_{1},...,i_{k})-1}\,,\;I_{i_{1},..,i_{t}+1,..,i_{k},j+\ell_{t}(i_{1},...,i_{k})}}\hskip
8.5359pt\text{and}\hskip
8.5359pt\varphi_{I_{i_{1},...,i_{k},j-1},\,I_{i_{1},...,i_{k},j}}^{I_{i_{1},...,i_{k},j+r_{s}(i_{1},...,i_{k})-1}\,,\;I_{i_{1},...,i_{k},j+r_{s}(i_{1},...,i_{k})}},$

for $t\in\\{1,...,k\\}$, and $s\in\\{1,...,d\\}$. Thus, by 1) in Lemma 3.3, we
have a group homomorphism $G\rightarrow\text{Diff}^{0}_{+}([0,1]).$ The main
technical step for proving Theorem A is the following proposition.

######  Proposition 3.4.

For an appropriate choice of length of the intervals
$|I_{i_{1},\ldots,i_{k},j}|$, the homeomorphisms
$f_{1},\ldots,f_{k},g_{1},\ldots,g_{d}$ are simultaneously diffeomorphisms of
class $C^{1+\alpha}$, for any $\alpha<1/k$.

The rest of Section 3.2 is devoted to give the proof of Proposition 3.4. We
will assume that $k\geq 2$ since, after Condition 3 in Proposition 3.1, we can
use the estimates from [6, §4] to ensure that, when $k=1$, the action is by
$C^{1+\alpha}$ diffeomorphisms for any $\alpha<1$.

So let $k\geq 2$ and consider $\alpha<1/k$. Choose positive real numbers
$p_{1},...,p_{k},r$ such that for all $n\in\\{1,...,k\\}$ they satisfy the
following conditions:

  1. I)

$\alpha+r\leq 2$,

  2. II)

$d(r-1)\leq(1-\alpha)$,

  3. III)

$2dr\leq p_{n}$,

  4. IV)

$2d\leq p_{n}(1-\alpha)$,

  5. V)

$\frac{1}{p_{1}}+\cdots+\frac{1}{p_{k}}+\frac{1}{r}<1$,

  6. VI)

$\alpha\leq\frac{1}{p_{n}}+\frac{1}{r}$ and $\alpha\leq\frac{r}{p_{n}(r-1)}$.

For instance, one can take $p_{1}=\cdots=p_{k}=3d/\alpha$ and $r=3d/(3d-1)$.

Now define the lengths of the intervals $I_{i_{1},\ldots,i_{k},j}$ as

$\lvert I_{i_{1},\ldots,i_{k},j}\rvert=\frac{1}{\lvert
i_{1}\rvert^{p_{1}}+\cdots+\lvert i_{k}\rvert^{p_{k}}+\lvert j\rvert^{r}+1}.$

From condition V) it follows that $\sum\lvert
I_{i_{1},\ldots,i_{k},j}\rvert<\infty$, hence this family of intervals can be
disposed on a finite interval respecting the lexicographic order. After
renormalization, we can assume that this interval is $[0,1]$.

Following [14], we say that two real-valued functions $f$ and $g$ satisfy
$f\prec g$ if there is a constant $M>0$ such that $\lvert f(x)\rvert\leqslant
Mg(x)$ for all $x$. We also write $f\asymp g$ if $f\prec g$ and $g\prec f$.

Let $\theta$ be a $C^{2}$ real-valued function satisfying
$\theta(\xi)=\lvert\xi\rvert^{r}$ for $\lvert\xi\rvert\geqslant 1$, and
$\theta(0)=0$. Consider the auxiliary functions ($C^{2}$ with respect to
$\xi$):

  * •

$\psi(i_{1},\ldots,i_{k},\xi):=1+\lvert i_{1}\rvert^{p_{1}}+\cdots+\lvert
i_{k}\rvert^{p_{k}}+\theta(\xi),$

  * •

$\Psi_{i_{1},\ldots,i_{k}}(\xi):=\text{log}(\psi(i_{1},\ldots,i_{k},\xi)).$

######  Lemma 3.5.

Let $S=1+\lvert i_{1}\rvert^{p_{1}}+\cdots+\lvert i_{k}\rvert^{p_{k}}$, and
suppose $\lvert\xi-j\rvert\leq C(S^{1/r}+(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d})$ for some positive constant $C$. Then

$\psi(i_{1},\ldots,i_{k},j)\asymp\psi(i_{1},\ldots,i_{k},\xi).$

Proof. By symmetry, it is enough to show that
$\frac{\psi(i_{1},\ldots,i_{k},\xi)}{\psi(i_{1},\ldots,i_{k},j)}$ is bounded
above. For this we note that

$\displaystyle\frac{\psi(i_{1},\ldots,i_{k},\xi)}{\psi(i_{1},\ldots,i_{k},j)}$
$\displaystyle\prec\frac{1+\lvert i_{1}\rvert^{p_{1}}+\cdots+\lvert
i_{k}\rvert^{p_{k}}+\lvert j\rvert^{r}+\lvert\xi-j\rvert^{r}}{1+\lvert
i_{1}\rvert^{p_{1}}+\cdots+\lvert i_{k}\rvert^{p_{k}}+\lvert j\rvert^{r}}$
$\displaystyle\prec 1+\frac{S+(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{dr}}{\psi(i_{1},\ldots,i_{k},j)}$ $\displaystyle\prec
2+\frac{(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{dr}}{\psi(i_{1},\ldots,i_{k},j)}.$

Where we repeatedly use the inequality $\lvert x+y\rvert^{a}\prec\lvert
x\rvert^{a}+\lvert y\rvert^{a}$, which holds for any $a>0$. Now just notice
that the last expression is bounded. Indeed, since $(\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{dr}\prec\lvert
i_{1}\rvert^{dr}+\cdots+\lvert i_{k}\rvert^{dr}$, it is enough to observe that
for each $n\in\\{1,\ldots,k\\}$,

$\lvert
i_{n}\rvert^{dr}\leq(\psi(i_{1},\ldots,i_{k},j))^{\frac{dr}{p_{n}}}\leq\psi(i_{1},\ldots,i_{k},j),$

which holds thanks to condition III). $\hfill\square$

####  3.2.1 The maps $g_{s}$ are $C^{1+\alpha}$-diffeomorphisms

We start the proof of Proposition 3.4 by showing that the map $g_{s}$, for
$s\in\\{1,\ldots,d\\}$, are of class $C^{1+\alpha}$. That is, we want to show
that there is constant $C>0$ such that

$\frac{\lvert\textup{log}Dg_{s}(x)-\textup{log}Dg_{s}(y)\rvert}{\lvert
x-y\rvert^{\alpha}}\leq C\text{ for all $x,y\in[0,1]$ }.$

To check this, it is enough to consider points $x,y$ in intervals
$I_{i_{1},\ldots,i_{k},j}$ and $I_{i_{1},\ldots,i_{k},j^{\prime}}$. Indeed,
after condition 2) in Lemma 3.3 and the definition of $g_{s}$, it follows that
$g_{s}$ has derivative 1 at the end points of the intervals
$\cup_{j}I_{i_{1},\ldots,i_{k},j}$, so this case follows by applying
triangular inequality.

Case 1: The points $x,y$ belongs to the same $I:=I_{i_{1},\ldots,i_{k},j}.$

Condition 3) in Lemma 3.3 provides a Lipchitz constant for $\log(Dg_{s})$. So
it is enough to bound

$\frac{1}{\lvert I\rvert^{\alpha}}\left|\frac{\lvert I\rvert\lvert
J^{\prime}\rvert}{\lvert J\rvert\lvert I^{\prime}\rvert}-1\right|,$

where $I^{\prime}=I_{i_{1},\ldots,i_{k},j-1}$,
$J=I_{i_{1},\ldots,i_{k},j+r_{s}(i_{1},\ldots,i_{d})}$ and
$J^{\prime}=I_{i_{1},\ldots,i_{k},j+r_{s}(i_{1},\ldots,i_{k})-1}$.

We will in fact bound the following asymptotically equivalent expression

$\frac{1}{\lvert I\rvert^{\alpha}}\text{log}\frac{\lvert I\rvert\lvert
J^{\prime}\rvert}{\lvert J\rvert\lvert I^{\prime}\rvert}.$

For this notice that $\text{log}\frac{\lvert I\rvert\lvert
J^{\prime}\rvert}{\lvert J\rvert\lvert I^{\prime}\rvert}$ equals to

$\Psi_{i_{1},\ldots,i_{k}}(j+r_{s}(i_{1},\ldots,i_{k}))-\Psi_{i_{1},\ldots,i_{k}}(j+r_{s}(i_{1},\ldots,i_{k})-1)-(\Psi_{i_{1},\ldots,i_{k}}(j)-\Psi_{i_{1},\ldots,i_{k}}(j-1)).$

So applying twice the Mean Value Theorem to the function
$x\mapsto\Psi_{i_{1},\ldots,i_{k}}(j+1+x)-\Psi_{i_{1},\ldots,i_{k}}(j+x),$ we
have

$\left\lvert\text{log}\frac{\lvert I\rvert\lvert J^{\prime}\rvert}{\lvert
J\rvert\lvert I^{\prime}\rvert}\right\rvert=\lvert
r_{s}(i_{1},\ldots,i_{k})\rvert\lvert
D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)\rvert,$ (5)

where $\xi$ is a point in the convex hull of $\\{j-1,j,j-1+r_{s},j+r_{s}\\}$.
Let us find an upper bound for $\lvert
D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)\rvert$. Since $D\theta$ and
$D^{2}\theta$ are bounded in $[-1,1]$, and

$D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)=\frac{D^{2}\theta(\xi)}{\psi(i_{1},\ldots,i_{k},\xi)}-\frac{(D\theta(\xi))^{2}}{(\psi(i_{1},\ldots,i_{k},\xi))^{2}},$

we have that

$D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)\prec\frac{1}{\psi(i_{1},\ldots,i_{k},\xi)}$

for all $\xi\in[-1,1]$. On the other hand for $\xi\notin[-1,1]$ we have that
$\theta(\xi)=\lvert\xi\rvert^{r}$. So, observing $\lvert\xi\rvert^{r-2}<1$ and
$\lvert\xi\rvert^{r}/\psi(i_{1},\ldots,i_{k},\xi)<1$, it follows that

$D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)\prec\frac{\lvert\xi\rvert^{r-2}}{\psi(i_{1},\ldots,i_{k},\xi)}\prec\frac{1}{\psi(i_{1},\ldots,i_{k},\xi)}.$
(6)

Now going back to equation (5) and using 3) of Proposition 3.1 , we have

$\text{log}\frac{\lvert I\rvert\lvert J^{\prime}\rvert}{\lvert J\rvert\lvert
I^{\prime}\rvert}\prec\frac{\lvert i_{1}\rvert^{d}+\cdots+\lvert
i_{k}\rvert^{d}}{\psi(i_{1},\ldots,i_{k},\xi)}.$

Note that for all $n\in\\{1,\ldots,k\\}$ the condition IV) yields

$\lvert
i_{n}\rvert^{d}\leq(\psi(i_{1},\ldots,i_{k},\xi))^{\frac{d}{p_{n}}}\leq(\psi(i_{1},\ldots,i_{k},\xi))^{(1-\alpha)}.$

Finally by Lemma 3.5 we conclude

$\frac{1}{\lvert I\rvert^{\alpha}}\text{log}\frac{\lvert I\rvert\lvert
J^{\prime}\rvert}{\lvert J\rvert\lvert
I^{\prime}\rvert}\prec\frac{(\psi(i_{1},\ldots,i_{k},\xi))^{-\alpha}}{\lvert
I\rvert^{\alpha}}\prec\frac{(\psi(i_{1},\ldots,i_{k},j))^{-\alpha}}{\lvert
I\rvert^{\alpha}}=1,$

as desired.

Case 2: The point $x$ belong to $I_{i_{1},\ldots,i_{k},j}$ and $y$ belong to
$I_{i_{1},\ldots,i_{k},j^{\prime}}.$

We assume without loss of generality that $j<j^{\prime}$. The condition 4) of
Lemma 3.3 tells us that
$\lvert\textup{log}Dg_{s}(x)-\textup{log}Dg_{s}(y)\rvert$ is bounded above by

$\left\lvert\text{log}\frac{\lvert I_{i_{1},\ldots,i_{k},j+r_{s}}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}+r_{s}}\rvert}}\right\rvert+\left\lvert\text{log}\frac{\lvert
I_{i_{1},\ldots,i_{k},j+r_{s}-1}\rvert\lvert
I_{i_{1},\ldots,i_{k},j}\rvert}{{\lvert I_{i_{1},\ldots,i_{k},j-1}\rvert\lvert
I_{i_{1},\ldots,i_{k},j+r_{s}}\rvert}}\right\rvert+\left\lvert\text{log}\frac{\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}+r_{s}-1}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}-1}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}+r_{s}}\rvert}}\right\rvert.$

The estimates in Case 1 allow us to estimate the last two terms (divided by
$|x-y|^{\alpha}$), thus we only need to blound the first term. So we look for
a uniform bound for

$\frac{1}{\lvert x-y\rvert^{\alpha}}\left\lvert\text{log}\frac{\lvert
I\rvert\lvert J^{\prime}\rvert}{{\lvert I^{\prime}\rvert\lvert
J\rvert}}\right\rvert,$ (7)

where $I=I_{i_{1},\ldots,i_{k},j}$,
$I^{\prime}=I_{i_{1},\ldots,i_{k},j^{\prime}}$,
$J=I_{i_{1},\ldots,i_{k},j+r_{s}}$ and
$J^{\prime}=I_{i_{1},\ldots,i_{k},j^{\prime}+r_{s}}$. Assume that $j$,
$j^{\prime}$ are positive (the case where both are negative follows by
symmetry, and if they have different sign, it suffices to consider an
intermediate comparison with the term corresponding to $j^{\prime\prime}=0$).
Assume further that $j^{\prime}-j\geq 2$ (the case where $j^{\prime}-j=1$
follows from the previous one, passing through the point that separates the
intervals and using triangular inequality). Again, applying twice the Mean
Value Theorem to the function

$x\mapsto\Psi_{i_{1},\ldots,i_{k}}(j-r_{s}+x)-\Psi_{i_{1},\ldots,i_{k}}(j+x)$

yields

$\left\lvert\text{log}\frac{\lvert I\rvert\lvert J^{\prime}\rvert}{{\lvert
I^{\prime}\rvert\lvert J\rvert}}\right\rvert=\lvert j-j^{\prime}\rvert\lvert
r_{s}(i_{1},\ldots,i_{k})\rvert\lvert
D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)\rvert,$ (8)

for a certain $\xi$ in the convex hull of
$\\{j,j^{\prime},j+r_{s},j^{\prime}+r_{s}\\}$.

We start by bounding $|x-y|^{-\alpha}$. For this note that by Case 1 and the
triangle inequality, we can (and will) assume that $x$ is the left endpoint of
$I$ and $y$ is the right endpoint of $I^{\prime}$. This yields

$\frac{1}{\lvert
x-y\rvert^{\alpha}}=\left(\frac{1}{\sum_{\ell=j}^{j^{\prime}}\lvert
I_{i_{1},\ldots,i_{k},\ell}\rvert}\right)^{\alpha}\leq\left(\frac{1}{\lvert
j-j^{\prime}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}\right)^{\alpha},$

where the last inequality holds because $\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert<\lvert
I_{i_{1},\ldots,i_{k},\ell}\rvert$ for $\ell<j^{\prime}$. Note that if in
addition $\lvert j^{\prime}-j\rvert\leq C(S^{1/r}+(\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d})$, for some $C>0$, we can use Lemma
3.5 to compare $\lvert I\rvert$ with $\lvert I^{\prime}\rvert$, and eventually
obtain the inequality

$\frac{1}{\lvert x-y\rvert^{\alpha}}\prec\left(\frac{1}{\lvert
j-j^{\prime}\rvert\lvert I_{i_{1},\ldots,i_{k},j}\rvert}\right)^{\alpha}.$ (9)

We now exhibit a bound for (7). We consider three separate cases. Let $M$ be
the constant in Proposition 3.1 and let
$S=1+|i_{1}|^{p_{1}}+\cdots+|i_{k}|^{p_{k}}$.

  1. $i)$

The integers $j,j^{\prime}$ belongs to $[0,2M(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d}]$. Since
$\xi\in\text{conv}\\{j,j^{\prime},j+r_{s},j^{\prime}+r_{s}\\}$ it follows from
Lemma 3.5 that

$\lvert
D^{2}(\Psi_{i_{1},\ldots,i_{k}})(\xi)\rvert\prec\frac{1}{\psi(i_{1},\ldots,i_{k},\xi)}\asymp\frac{1}{\psi(i_{1},\ldots,i_{k},j)}.$

Furthermore, we have that

$\lvert j-j^{\prime}\rvert\lvert r_{s}(i_{1},\ldots,i_{k})\rvert\prec(\lvert
i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{2d}\prec(\psi(i_{1},\ldots,i_{k},j))^{1-\alpha},$

where the last inequality holds from condiition IV). If we combine this with
(8), (9), we conclude that

$\frac{1}{\lvert x-y\rvert^{\alpha}}\left\lvert\text{log}\frac{\lvert
I\rvert\lvert J^{\prime}\rvert}{{\lvert I^{\prime}\rvert\lvert
J\rvert}}\right\rvert\prec\frac{1}{\lvert
I\rvert^{\alpha}}\frac{(\psi(i_{1},\ldots,i_{k},j))^{1-\alpha}}{\psi(i_{1},\ldots,i_{k},j)}=1.$

  2. $ii)$

The integer $j,j^{\prime}$ belongs to222the constant $k^{d}$ is just to ensure
that the interval is non-empty.  $[2M(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d},2Mk^{d}S^{1/r}]$. Similarly to $i)$, the reader can check
that we are in the hypotheses of Lemma 3.5 and that $\lvert\xi\rvert\geq
M(\lvert i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d}$. Therefore, by (6), (8)
and (9), we get

$\displaystyle\frac{1}{\lvert
x-y\rvert^{\alpha}}\left\lvert\text{log}\frac{\lvert I\rvert\lvert
J^{\prime}\rvert}{{\lvert I^{\prime}\rvert\lvert J\rvert}}\right\rvert$
$\displaystyle\prec\left(\frac{1}{\lvert j-j^{\prime}\rvert\lvert
I_{i_{1},\ldots,i_{k},j}\rvert}\right)^{\alpha}\lvert
j^{\prime}-j\rvert\frac{(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d}\lvert\xi\rvert^{r-2}}{\psi(i_{1},\ldots,i_{k},\xi)}$
$\displaystyle\prec\lvert j^{\prime}-j\rvert^{1-\alpha}\frac{(\lvert
i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d(r-1)}}{\psi(i_{1},\ldots,i_{k},j)^{1-\alpha}}.$

To prove that this last expression is bounded, it is enough to show that

$\lvert j^{\prime}-j\rvert^{1-\alpha}(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d(r-1)}\prec\psi(i_{1},\ldots,i_{k},j)^{1-\alpha}.$

Since $j^{\prime}-j\leq 2Mk^{d}S^{1/r}$, it follows that

$\lvert j^{\prime}-j\rvert^{1-\alpha}(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d(r-1)}\prec(1+\lvert i_{1}\rvert^{p_{1}}+\cdots+\lvert
i_{k}\rvert^{p_{k}})^{\frac{(1-\alpha)}{r}}(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d(r-1)},$

so it suffices to prove that, given $n,m\in\\{1,...,k\\}$ we have

$\lvert i_{n}\rvert^{\frac{p_{n}(1-\alpha)}{r}}\lvert
i_{m}\rvert^{d(r-1)}\prec(\psi(i_{1},...,i_{k},j))^{1-\alpha}.$ (10)

But note that

$\lvert i_{n}\rvert^{\frac{p_{n}(1-\alpha)}{r}}\lvert
i_{m}\rvert^{d(r-1)}\leq(\psi(i_{1},\ldots,i_{k},j))^{\frac{(1-\alpha)}{r}+\frac{d(r-1)}{p_{m}}},$

and that conditions II) and V) guarantee
$\frac{(1-\alpha)}{r}+\frac{d(r-1)}{p_{m}}\leq(1-\alpha)$, which implies (10).

  3. $iii)$

Finally suppose that the integers $j,j^{\prime}$ belongs to
$[2Mk^{d}S^{1/r},\infty]$.

If $j^{\prime}\leq 2j$, then

$\frac{\psi(i_{1},\ldots,i_{k},j^{\prime})}{\psi(i_{1},\ldots,i_{k},j)}\prec
1+\frac{\lvert j-j^{\prime}\rvert^{r}}{\psi(i_{1},\ldots,i_{k},j)}\prec
1+\frac{\lvert j\rvert^{r}}{\psi(i_{1},\ldots,i_{k},j)}\leq 2.$ (11)

In particular, the intervals $\lvert I^{\prime}\rvert$ and $\lvert I\rvert$
have comparable size and hence we conclude that (9) still holds. Also note
that $j^{\prime}\leq 2j$ implies $\lvert\xi-j\rvert\leq\lvert j\rvert+M(\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d}$. Then, proceeding as in $ii)$, we
have that

$\frac{1}{\lvert x-y\rvert^{\alpha}}\left\lvert\text{log}\frac{\lvert
I\rvert\lvert J^{\prime}\rvert}{\lvert I^{\prime}\rvert\lvert
J\rvert}\right\rvert\prec\frac{\lvert j\rvert^{1-\alpha}(\lvert
i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d(r-1)}}{\psi(i_{1},\ldots,i_{k},j)^{1-\alpha}}$

The reader can check, again as in $ii)$, that this last expression is bounded.

For the case $j^{\prime}>2j$. We have

$\displaystyle\lvert x-y\rvert$
$\displaystyle=\sum_{\ell=j}^{j^{\prime}}\lvert
I_{i_{1},\ldots,i_{k},\ell}\rvert=\sum_{\ell=j}^{j^{\prime}}\frac{1}{\lvert
i_{1}\rvert^{p_{1}}+\cdots+\lvert i_{k}\rvert^{p_{k}}+\lvert\ell\rvert^{r}}$
$\displaystyle\succ\sum_{\ell=j}^{j^{\prime}}\frac{1}{\lvert\ell\rvert^{r}}\succ\int_{\ell=j}^{j^{\prime}}\frac{1}{x^{r}}dx\succ\frac{1}{\lvert
j\rvert^{r-1}},$

where the last inequality holds because $j^{\prime}>2j$. On the other hand,
applying the Mean Value Theorem, it follows that

$\displaystyle\text{log}\frac{\lvert I\rvert\lvert J^{\prime}\rvert}{\lvert
I^{\prime}\rvert\lvert J\rvert}$ $\displaystyle=\lvert
r_{s}(i_{1},\ldots,i_{k})\rvert\lvert
D(\Psi_{i_{1},\ldots,i_{k}})(\xi)-D(\Psi_{i_{1},\ldots,i_{k}})(\tilde{\xi})\rvert,$

with $\xi\in\text{conv}\\{j,j+r_{s}\\}$ and
$\tilde{\xi}\in\text{conv}\\{j^{\prime},j^{\prime}+r_{s}\\}$. Therefore,
observing that the function $\xi\mapsto
D(\Psi_{i_{1},\ldots,i_{k}})(\xi)=r\lvert\xi\rvert^{r-1}/\psi(i_{1},\ldots,i_{k},\xi)$
is decreasing, we have

$\frac{1}{\lvert x-y\rvert^{\alpha}}\left\lvert\text{log}\frac{\lvert
I\rvert\lvert J^{\prime}\rvert}{\lvert I^{\prime}\rvert\lvert
J\rvert}\right\rvert\prec(\lvert i_{1}\rvert+\cdots+\lvert
i_{k}\rvert)^{d}\frac{\lvert
j\rvert^{(\alpha+1)(r-1)}}{\psi(i_{1},\ldots,i_{k},j)}.$

Now we want to see that this last expression is bounded, in other words that
the inequality $\lvert j\rvert^{(\alpha+1)(r-1)}(\lvert
i_{1}\rvert+\cdots+\lvert i_{k}\rvert)^{d}\prec\psi(i_{1},\ldots,i_{k},j)$
holds. For this, arguing as in (10), it is enough to check that for all
$n\in\\{1,...,k\\}$ the inequality

$\frac{(\alpha+1)(r-1)}{r}+\frac{d}{p_{n}}\leq 1$

holds. To see this, note that from IV) it follows that
$\frac{d}{p_{n}}\leq\frac{1-\alpha}{2}$. Finally notice that
$\frac{(\alpha+1)(r-1)}{r}+\frac{1-\alpha}{2}\leq 1\Leftrightarrow r\leq 2$,
which is ensured by condition I).

####  3.2.2  The maps $f_{t}$ are $C^{1+\alpha}$-diffeomorphisms

In the same way that for the maps $g_{s}$, we want to see that for all
$x,y\in[0,1]$

$\frac{\lvert Df_{t}(x)-Df_{t}(y)\rvert}{\lvert x-y\rvert^{\alpha}}\leqslant
C\text{ holds for some constant }C>0.$

To simplify notation we will only work with $t=1$ as the other cases are
analogous. As for the case of the maps $g_{s}$, we only have two cases to
analyze.

Case 1: The points $x,y$ belongs to the same interval
$I_{i_{1},\ldots,i_{k},j}$.

By Lemma 3.3 it is enough to show that the following expression is uniformly
bounded

$\frac{1}{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert^{\alpha}}\text{log}\frac{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j+\ell_{1}-1}\rvert}{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j+\ell_{1}}\rvert\lvert
I_{i_{1},\ldots,i_{k},j-1}\rvert}.$

To see the this, simply note that the above expression is equal to

$\frac{1}{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert^{\alpha}}\text{log}\frac{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j-1}\rvert}{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j}\rvert\lvert
I_{i_{1},\ldots,i_{k},j-1}\rvert}+\frac{1}{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert^{\alpha}}\text{log}\frac{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j+\ell_{1}-1}\rvert}{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j+\ell_{1}}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j-1}\rvert}.$

By condition VI) we know from [6, §3.3] that the first term is uniformly
bounded. The second term is bounded as well since it is the same that we bound
when dealing with $g_{s}$ (changing $i_{1}$ for $i_{1}+1$).

Case 2: The point $x\in I=I_{i_{1},\ldots,i_{k},j}$ and $y\in
J=I_{i_{1},\ldots,i_{k},j^{\prime}}$, with $j<j^{\prime}$.

Here we can use 4) from Lemma 3.3 to bound $\lvert\log Df_{1}(x)-\log
Df_{1}(y)\rvert$ by

$\left\lvert\text{log}\frac{\lvert
I_{i_{1}+1,\ldots,i_{k},j+\ell_{1}}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert\lvert
I_{i_{1}+1,\ldots,i_{k},j^{\prime}+\ell_{1}}\rvert}}\right\rvert+\left\lvert\text{log}\frac{\lvert
I_{i_{1}+1,\ldots,i_{k},j+\ell_{1}-1}\rvert\lvert
I_{i_{1},\ldots,i_{k},j}\rvert}{{\lvert I_{i_{1},\ldots,i_{k},j-1}\rvert\lvert
I_{i_{1}+1,\ldots,i_{k},j+\ell_{1}}\rvert}}\right\rvert+\left\lvert\text{log}\frac{\lvert
I_{i_{1}+1,\ldots,i_{k},j^{\prime}+\ell_{1}-1}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}-1}\rvert\lvert
I_{i_{1}+1,\ldots,i_{k},j^{\prime}+\ell_{1}}\rvert}}\right\rvert,$

and then work in the same way as for the functions $g_{s}$. For example, we
express the term

$\frac{1}{\lvert x-y\rvert^{\alpha}}\text{log}\frac{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j+\ell_{1}}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1},\ldots,i_{k},j}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j^{\prime}+\ell_{1}}\rvert}}$

as

$\frac{1}{\lvert x-y\rvert^{\alpha}}\text{log}\frac{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j}\rvert\lvert
I_{i_{1},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j^{\prime}}\rvert\lvert
I_{i_{1},\ldots,i_{k},j}\rvert}}+\frac{1}{\lvert
x-y\rvert^{\alpha}}\text{log}\frac{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j+\ell_{1}}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j^{\prime}}\rvert}{{\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j}\rvert\lvert
I_{i_{1}+1,i_{2},\ldots,i_{k},j^{\prime}+\ell_{1}}\rvert}}.$

The first term is bounded by [6, §3.3] and the second is also bounded by the
same argument used for the functions $g_{s}$.

###  3.3 Faithful action

Given $s\in\\{1,\ldots,d\\}$ and a compact interval $I_{s}$, we have seen how
to produce certain actions

$\phi_{s}:G\rightarrow\text{Diff}_{+}^{1+\alpha}(I_{s}),$

where the subgroup $\langle g_{1},\ldots,g_{s-1}\rangle$ acts trivial, while
$\langle g_{s}\rangle$ acts faithful (recall that we are using the generating
set from Lemma 2.3).

To obtain a faithful action we do the following: consider $I_{1},\ldots,I_{d}$
compact intervals such that for all $s\in\\{1,\ldots,d-1\\}$, $I_{s+1}$ is
contiguous to $I_{s}$ by the right. Then define over $I:=I_{1}\cup\cdots\cup
I_{d}$ the action $\phi:G\rightarrow\text{Diff}_{+}^{1+\alpha}(I)$ as

$\phi\mid_{I_{s}}=\phi_{s}.$

We claim that $\phi$ is injective. Indeed, since $Z(G)\leqslant A$, by the
Proposition 2.1 we only need to check that $\phi\mid_{A}$ is inyective. Let
$g\in A$ be an element that acts trivially on $I$. Using the generator of $A$,
we have that there exist $j_{1},\ldots,j_{d}\in\mathbb{Z}$ such that
$g=g_{1}^{j_{1}}\cdots g_{d}^{j_{d}}$. Now, since $\phi(g)=id$, it follows
that

$\phi_{s}(g)=id,\;\forall s\in\\{1,\ldots,d\\}.$

This yields that $j_{d}=\cdots=j_{1}=0$ and hence $g$ is the trivial element.
This finishes the proof of Theorem A.

##  4 Examples

In this section we give examples of nilpotent groups for which we can compute
their critical regularity. In each case we use Theorem A to obtain a lower
bound for the critical regularity and we argue that in our examples this is
also an upper bound for the regularity.

We begin by recalling that if $G$ is a finitely-generated nilpotent group of
homeomorphisms of $(0,1)$ that has no global fixed points (or more generally a
group acting without crossings, see [20, 10]), then there is a well-defined
group homomorphism $\rho:G\rightarrow\mathbb{R},$ which is usually called the
translation number of the action. This map characterizes the elements of $G$
that have fixed points, in the sense that $\rho(g)=0$ if and only if $g$ has a
fixed point in $(0,1)$. We further note that the action of $G$ on the interval
has no crossings. By this we mean that if an element $f\in G$ fixes an open
subinterval $I$ of $(0,1)$ and satisfies that $f(x)\neq x\text{ for all
}x\text{ in }I$, then for any other $g\in G$ we have that $g(I)=I$ or
$g(I)\cap I=\emptyset$. See [20, §2.2.5] for background. With this, it is easy
to prove the following result that we will repeatedly use.

######  Lemma 4.1.

Let $G\leqslant\textup{Diff}_{+}^{0}(0,1)$ be a nilpotent group, and let $c\in
G$ be a non-trivial element such that $c=[a,b]$ for some elements $a,b\in G$.
If $c$ fix an open interval $I$ and has no fixed point inside, then either $a$
or $b$ move $I$ (disjointly).

Proof. Looking for a contradiction, assume that $a$ and $b$ fix $I$. Then we
have the translation number homomorphism for the group $\langle
a,b,c\rangle\leqslant\text{Diff}_{+}^{0}(I)$. Since $c$ is a commutator, it is
in the kernel of this morphism. Hence we conclude that $c$ has a fixed point
inside $I$, which is contrary to our assumptions. $\hfill\square$

To obtain upper bounds for the regularity of our groups we will use two
results from B. Deroin, V. Kleptsyn and A. Navas [8].

######  Theorem 4.2.

Let $f_{1},\ldots,f_{k}$ be $C^{1}$-diffeomorphisms of the interval $[0,1]$
that commute with a $C^{1}$-diffeomorphism $g$. Assume that $g$ fixes a
subinterval $I$ of $[0,1]$ and its restriction to $I$ is non-trivial. Assume
moreover that for a certain $0<\alpha<1$ and a sequence of indexes
$i_{j}\in\\{1,\ldots,k\\}$, the sum

$\sum_{j\geq 0}\lvert f_{i_{j}}\cdots f_{i_{1}}(I)\rvert^{\alpha}<\infty.$

Then $f_{1},\ldots,f_{k}$ cannot be all of class $C^{1+\alpha}$.

The following lemma is useful to get into the hypotheses of Theorem 4.2.
Although it is stated in a slightly different way, the reader can check that
the proof is exactly the same as that of the [8, Lemma 3.3].

######  Lemma 4.3.

Let $f_{1},\ldots,f_{k}$ be $C^{1}$-diffeomorphisms of $[0,1]$, and $I$
subinterval of $[0,1]$ such that $\mathbb{Z}^{k}\simeq\langle
f_{1},\ldots,f_{k}\rangle/\textup{Stab}(I)$, where $\textup{Stab}(I)$ is the
stabilizer of $I$ (which is assumed to be a normal subgroup). Then, if
$\alpha>1/k$ there exist a sequence $(f_{i_{j}})_{j\in\mathbb{N}}$ of elements
in $\\{f_{1},\ldots,f_{k}\\}$ such that

$\sum_{j\geq 0}\lvert f_{i_{j}}\cdots f_{i_{1}}(I)\rvert^{\alpha}<\infty.$

###  4.1 Heisenberg Groups

For a natural number $n\geq 1$, the discrete $(2n+1)$-dimensional _Heisenberg
group_ , is defined as the set of matrices

$\mathscr{H}_{n}:=\left\\{\begin{pmatrix}1&\vec{x}&c\\\
\vec{0}^{t}&I_{n}&\vec{y}^{\,t}\\\
0&\vec{0}&1\end{pmatrix}:\vec{x},\vec{y}\in\mathbb{Z}^{n},c\in\mathbb{Z}\text{
and }I_{n}\text{ is the identity matrix of size }n\right\\},$

with the usual matrix product. Note that the center of $\mathscr{H}_{n}$
coincides with the commutator subgroup and is generated by the matrix

$\textbf{C}:=\begin{pmatrix}1&\vec{0}&1\\\ \vec{0}^{t}&I_{n}&\vec{0}^{\,t}\\\
0&\vec{0}&1\end{pmatrix}.$

We want to prove Theorem B, before this, it will be useful for us to bound the
rank of maximal abelian subgroups. Assume that there exists a maximal abelian
subgroup of $\mathscr{H}_{n}$ (note that it contains C) of rank $m$. Then we
can choose elements

$\textbf{A}_{i}:=\begin{pmatrix}1&\vec{a_{i}}&c_{i}\\\
\vec{0}^{t}&I_{n}&\vec{b_{i}}^{t}\\\
0&\vec{0}&1\end{pmatrix}\in\mathscr{H}_{n}\hskip 5.69046pt\text{for}\hskip
5.69046pti\in\\{1,\ldots,m-1\\},$

such that
$\langle\textbf{A}_{1},\ldots\textbf{A}_{m-1},\textbf{C}\rangle\simeq\mathbb{Z}^{m}.$
Note that the commutativity of these matrices is equivalent to the equations

$\vec{a_{i}}\cdot\vec{b_{j}}=\vec{a_{j}}\cdot\vec{b_{i}}\hskip
8.5359pt\forall\;i,j\in\\{1,...,m-1\\}.$ (12)

Note also that $\\{\textbf{A}_{1},\ldots,\textbf{A}_{m-1},\textbf{C}\\}$
generates a free abelian subgroup of rank $m$ if and only if the set of
vectors
$\mathscr{B}:=\\{(\vec{b}_{i},\vec{a}_{i})\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}:1\leq
i\leq m-1\\}$ is linearly independent over $\mathbb{Z}$. Indeed, if we have a
dependency relation, say
$r(\vec{b}_{1},\vec{a}_{1})\in\langle(\vec{b}_{2},\vec{a}_{2}),\ldots,(\vec{b}_{m-1},\vec{a}_{m-1})\rangle$
for some $0\neq r\in\mathbb{Z},$ then
$\textbf{A}_{1}^{r}\in\langle\textbf{A}_{2},\ldots,\textbf{A}_{m-1},\textbf{C}\rangle,$
which contradicts that the abelian group has rank $m$.

Having said this, we claim that $m\leq n+1$. To see this, note that, by
equations (12) any vector of the form $(\vec{a}_{i},-\vec{b}_{i})$, with
$1\leq i\leq m-1$, is perpendicular to $\langle\mathscr{B}\rangle$. Hence we
have two orthogonal subgroups of rank $m-1$, and thus $m-1\leq n$, which
proves our claim.

##### Realization.

Consider the abelian subgroup

$A:=\left\\{\begin{pmatrix}1&\vec{x}&c\\\ \vec{0}^{\,t}&I_{n}&\vec{0}^{\,t}\\\
0&\vec{0}&1\end{pmatrix}\;:\vec{x}\in\mathbb{Z}^{n}\text{ and
}c\in\mathbb{Z}\right\\},$

it has rank equal to $n+1$, which is the largest we can expect. Since the rank
of $\mathscr{H}_{n}/A$ is $n$, we have that Theorem A provides an injective
group homomorphism

$\mathscr{H}_{n}\hookrightarrow\text{Diff}_{+}^{1+\alpha}([0,1])\text{ for
}\alpha<1/n.$

##### Bounding the regularity.

Now we consider a faithful action
$\phi:\mathscr{H}_{n}\hookrightarrow\text{Diff}_{+}^{1}([0,1])$. Making a
little abuse of notation, we can think
$\mathscr{H}_{n}\leqslant\text{Diff}_{+}^{1}([0,1])$.

Since the commutator subgroup of $\mathscr{H}_{n}$ is generated by ${\bf C}$,
we deduce from Lemma 4.1 that C has fixed points inside $(0,1)$. Therefore, we
can find an interval $I\subsetneq[0,1]$ such that $\textbf{C}(I)=I\text{ and
}\textbf{C}(x)\neq x$ for all $x$ in the interior of $I$. Let $\text{Stab}(I)$
be the stabilizer of $I$. It is easy to see that this is an abelian subgroup.
Indeed, if we take $\textbf{A},\textbf{B}\in\text{Stab}(I)$ and assume that
they do not commute, then there must exist $m\in\mathbb{Z}$ such that
$[\textbf{A},\textbf{B}]=\textbf{C}^{m}$. Since C has no fixed points inside
$I$, Lemma 4.1 tell us that either A or B moves $I$ , which is a
contradiction. Note that Stab($I$) is a normal subgroup as it contains the
commutator subgroup.

Further, we know that there is a natural number $k$ and elements
$\textbf{B}_{1},\ldots,\textbf{B}_{k}\in\mathscr{H}_{n}$ such that

$\mathbb{Z}^{k}\simeq\frac{\mathscr{H}_{n}}{\text{Stab}(I)}=\frac{\langle\textbf{B}_{1},\ldots,\textbf{B}_{k}\rangle}{\text{Stab}(I)}.$

So, given $\alpha>1/k$, we can find by Lemma 4.3 a sequence
$(\textbf{B}_{i_{j}})_{j\in\mathbb{N}}$ of elements in
$\\{\textbf{B}_{1},\ldots,\textbf{B}_{k}\\}$ such that

$\sum_{j\geq
0}\lvert\textbf{B}_{i_{j}}\cdots\textbf{B}_{i_{1}}(I)\rvert^{\alpha}<\infty,$

and hence Theorem 4.2 yields that $\phi$ is not an action by
$C^{1+\alpha}$-diffeomorphisms.

Now since the rank of $\text{Stab}(I)$ is bounded by $n-1=\text{rank}(A)$, we
have that

$k=\text{rank}\left(\frac{\mathscr{H}_{n}}{\text{Stab}(I)}\right)\geq\text{rank}\left(\frac{\mathscr{H}_{n}}{A}\right)=n,$

which tells us that the regularity of the action $\phi$ is bounded by $1+1/n$.
So we conclude that

$\text{Crit}_{[0,1]}(\mathscr{H}_{n})=1+\frac{1}{n}.$

The reader can easily check the following corollary of the proof of Theorem B.

######  Corollary 4.4.

Let $G$ be a finitely-generated nilpotent group with cyclic center and such
that $[G,G]\leqslant Z(G)$. Then, the lower bound for $\textup{Crit}(G)$
obtain in Theorem A is also an upper bound.

###  4.2 Examples with large nilpotency degree

Theorem B gives us the critical regularity for the Heisenberg groups, which
are groups having nilpotency degree 2. In this subsection we provide more
examples of nilpotent groups where we can compute its critical regularity, but
their nilpotency degree can be arbitrarily large. As for the Heisenberg
groups, in these examples we will show that the lower bound provided by
Theorem A is also an upper bound.

Fix $d,k\in\mathbb{N}$, assume $d\geq k$ and consider a matrix $(m_{i,s})\in
M_{k}(\mathbb{Z})$ with non-zero determinant and positive entries. We let $G$
be the group generated by the set

$\\{g_{0}\\}\cup\\{g_{i,j}:(i,j)\in\\{1,\ldots,k\\}\times\\{1,\ldots,d\\}\\}\cup\\{f_{1},\ldots,f_{k}\\},$

subject to the relations

  * •

$[g_{0},g_{i,j}]=[g_{0},f_{i}]=[f_{s},f_{i}]=[g_{i,j},g_{l,m}]=e,$ $\forall$
$s,i,l\in\\{1,\ldots,k\\},\;j,m\in\\{1,\ldots,d\\}$,

  * •

$[f_{s},g_{i,j}]=g_{i,j-1}^{m_{i,s}}$ $\forall s,i\in\\{1,\ldots,k\\}$ y
$j\in\\{2,\ldots,d\\}$,

  * •

$[f_{s},g_{i,1}]=g_{0}^{m_{i,s}}$ $\forall s,i\in\\{1,\ldots,k\\}.$

Note that from the identities $[ab,c]=a[b,c]a^{-1}[a,c]$ and
$[a,bc]=[a,b]b[a,c]b^{-1}$, we immediately have the following additional
relations

  * •

$[f_{s}^{-1},g_{i,j}]\in\langle g_{0},g_{i,1},\ldots,g_{i,j-2}\rangle
g_{i,j-1}^{-m_{i,s}}\;\forall s,i\in\\{1,\ldots,k\\},j\in\\{2,\ldots,d\\}$,

  * •

$[f_{s}^{-1},g_{i,1}]=g_{0}^{-m_{i,s}}\;\forall s,i\in\\{1,\ldots,k\\}$.

It is easy to see that $G$ is a nilpotent group of degree $d+1$, and
$A=\langle\\{g_{0}\\}\cup\\{g_{i,j}:(i,j)\in\\{1,...,k\\}\times\\{1,\ldots,d\\}\\}\rangle$
is a maximal abelian subgroup containing the commutator of $G$ (see Lemma 4.5
below). Moreover $k$ is the torsion-free rank of $G/A$, therefore in view of
Theorem A we know that $G$ embeds in $\text{Diff}_{+}^{1+\alpha}([0,1])$ for
$\alpha<1/k$. To show that $1+1/k$ is actually an upper bound for the
regularity we are going to need the following elementary lemma.

######  Lemma 4.5.

For all $(i,j)\in\\{1,\ldots,k\\}\times\\{2,\ldots,d\\}$ and
$n_{1},\ldots,n_{k}\in\mathbb{Z}$ we have

  1. 1.

$[f_{1}^{n_{1}}\cdots f_{k}^{n_{k}},g_{i,j}]\in\langle
g_{0},g_{i,1},\ldots,g_{i,j-2}\rangle g_{i,j-1}^{\lambda_{i}},$

  2. 2.

$[f_{1}^{n_{1}}\cdots f_{k}^{n_{k}},g_{i,1}]=g_{0}^{\lambda_{i}},$

where $\lambda_{i}=\sum_{s=1}^{k}n_{s}m_{i,s}$. In particular, the subgroup
$A$ is a maximal abelian subgroup.

Proof. To show 1. we do induction on $n=\sum_{s=1}^{k}\lvert n_{s}\rvert$.

Note that when $n=1$ we have the result by the relations of $G$. So, consider
an arbitrary natural number $n=\sum_{j=1}^{k}\lvert n_{j}\rvert$ and assume
that $n_{k}<0$ (the other case is similar). For all $i\in\\{1,\ldots,k\\}$ and
$j\in\\{2,...,d\\}$ we have that

$[f_{1}^{n_{1}}\cdots f_{k}^{n_{k}},g_{i,j}]=[f_{1}^{n_{1}}\cdots
f_{k}^{n_{k}+1},[f_{k}^{-1},g_{i,j}]]\;[f_{k}^{-1},g_{i,j}]\;[f_{1}^{n_{1}}\cdots
f_{k}^{n_{k}+1},g_{i,j}],$

and since $[f_{k}^{-1},g_{i,j}]$ belongs to $\langle
g_{0},g_{i,1},\ldots,g_{i,j-2}\rangle g_{i,j-1}^{-m_{i,k}}$, it follows that
$[f_{1}^{n_{1}}\cdots f_{k}^{n_{k}+1},[f_{k}^{-1},g_{i,j}]]\in\langle
g_{0},g_{i,1},\ldots,g_{i,j-2}\rangle$. Also, by induction hypothesis we have

$[f_{1}^{n_{1}}\cdots f_{k}^{n_{k}+1},g_{i,j}]\in\langle
g_{0},g_{i,1},\ldots,g_{i,j-2}\rangle
g_{i,j-1}^{(\sum_{s=1}^{k-1}n_{s}m_{i,s}+(n_{k}+1)m_{i,k})}.$

Plugin these into the previous equation yields assertion 1. The proof of
assertion 2 is analogous.

$\hfill\square$

######  Remark 4.6.

The most useful part of Lemma 4.5 is the explicit expression for the integers
$\lambda_{i}$ appearing. These will be used in the proof of Theorem C.

##### Proof of Theorem C:

Suppose that $G$ embeds into $\text{Diff}_{+}^{1+\alpha}([0,1])$ for some
$\alpha>1/k$. Let $x_{0}$ be a point in $(0,1)$ such that $g_{0}(x_{0})\neq
x_{0}$ and define the intervals

$I_{0}:=(\underset{n}{\text{inf}}\;g_{0}^{n}(x_{0}),\underset{n}{\text{sup}}\;g_{0}^{n}(x_{0}))\text{
and
}I_{i,j}:=(\underset{n}{\text{inf}}\;g_{i,j}^{n}(x_{0}),\underset{n}{\text{sup}}\;g_{i,j}^{n}(x_{0})).$

Case 1: $f(I_{0})\cap I_{0}=\emptyset$ for all $f\in\langle
f_{1},\ldots,f_{k}\rangle\simeq\mathbb{Z}^{k}$.

In this case $I_{0}$ is a wandering interval for the dynamics of $\langle
f_{1},\ldots,f_{k}\rangle$. A contradiction is provided by Lemma 4.3 followed
by Theorem 4.2 since the central element $g_{0}$ acts non-trivially $I_{0}$.

Case 2: There is a non-trivial element $f\in\langle f_{1},\ldots,f_{k}\rangle$
such that $f(I_{0})=I_{0}$.

Let us put $f=f_{1}^{n_{1}}\cdots f_{k}^{n_{k}}$. Given
$i\in\\{1,\ldots,k\\}$, by the Lemma 4.5 we have that

$[f,g_{i,1}]=g_{0}^{\lambda_{i}}\text{ and }[f,g_{i,j}]\in\langle
g_{0},g_{i,1},\ldots,g_{i,j-2}\rangle g_{i,j-1}^{\lambda_{i}}\hskip
8.5359pt\text{ for all }j\in\\{2,\ldots,d\\},$ (13)

where $\lambda_{i}=\sum_{j=1}^{k}n_{j}m_{i,j}$. Since the vectors
$(m_{i,1},\ldots,m_{i,k})$ are linearly independent in $\mathbb{R}^{k}$, we
can choose $i$ to obtain $\lambda_{i}\neq 0$. Then, the relations (13) and
Lemma 4.1 implies that $g_{i,1}(I_{0})\cap I_{0}=\emptyset$. Since the action
has no crossings, the element $f$ also fixes the intervals $I_{i,j}$ and hence
the same argument also yields that $g_{i,j}(I_{i,j-1})\cap
I_{i,j-1}=\emptyset$ for all $j>2$. Therefore, $I_{0}$ is a wandering interval
for the action of $\langle g_{i,1},\ldots,g_{i,k}\rangle\simeq\mathbb{Z}^{k}$,
So, a contradiction is reached using Lemma 4.3 and Theorem 4.2 as before.
$\hfill\square$

###  4.3 An example with even higher regularity

It is easy to see that in some situations the regularity given by Theorem A is
not critical. In the examples that we know of, this is related to the fact
that the group can be splitted as a direct product of groups each of which
allows an embedding with better regularity. Take for example the groups of [6,
§4]. These are given by the presentation

$G_{d}:=\langle
f,g_{1},\ldots,g_{d}:[g_{i},g_{j}]=id,[f,g_{0}]=id,[f,g_{i}]=g_{i-1}\;\forall
j\geq 0,i\geq 1\rangle.$

Note that $G_{d}$ is isomorphic to a non-trivial semidirect product of the
form $\mathbb{Z}^{d}\rtimes\mathbb{Z}$. Now define the group $G:=G_{d}\times
G_{d}.$ On one hand, it is easy to see that

$G\simeq\mathbb{Z}^{2d}\rtimes\mathbb{Z}^{2},$

and $\mathbb{Z}^{2d}\times\\{0\\}$ is a maximal abelian subgroup of $G$.
Therefore, if we apply the Theorem A, we obtain an embedding of $G$ into
$\text{Diff}_{+}^{1+\alpha}([0,1])$ for all $\alpha<1/2$. However, on the
other hand, the critical regularity of $G$ es 2. Indeed, we can apply Theorem
A to each factor of $G$ to obtain an embedding of the factor into
$\text{Diff}_{+}^{1+\alpha}([0,1])$ for all $\alpha<1$. If we put these two
actions together acting on disjoint intervals (as we did in Section 3.3), we
end up with an embedding if $G$ into $\text{Diff}_{+}^{1+\alpha}([0,1])$ for
all $\alpha<1$.

## References

  * [1] H. Bass. The degree of polynomial growth of finitely generated nilpotent groups. Proc. Lond. Math. Soc. 25 (1972), 603-614.
  * [2] G. Bergman. Right-orderable groups that are not locally indicable. Pac. J. Math. 147 (1991), 243-248.
  * [3] C. Bonatti, I. Monteverde, A. Navas & C. Rivas. Rigidity for $C^{1}$ actions on the interval arising from hiperbolicity I: solvable groups. Math. Z. 286 (2017), 919-949.
  * [4] J. Brum, N. Matte Bon, C. Rivas & M. Triestino. Locally moving groups acting on the line and $\mathbb{R}$-focal actions. Arxiv:2104.14678 (2021).
  * [5] D. Calegari. Nonsmoothable, locally indicable groups actions on the interval. Alg. Geom. Top. 8 (2008), 609-613.
  * [6] G. Castro, E. Jorquera & A. Navas. Sharp regularity for certain nilpotent group actions on the interval. Math. Ann. 359 (2014), no. 1-2, 101-152.
  * [7] D. J. S. Robinson. A Course in the Theory of Groups. Number 80 in Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1996.
  * [8] B. Deroin, V. Kleptsyn & A. Navas. Sur la dynamique unidimensionnelle en régularité intermédiaire. Acta Math. 199 (2007), no. 2, 199-262.
  * [9] B. Deroin, V. Klepstyn, A. Navas & K. Parwani. Symmetric random walks on $\text{Homeo}_{+}(\mathbb{R})$. Ann. Probab. 41, 2066-2089 (May 2013).
  * [10] B. Deroin, A. Navas & C. Rivas. Groups, Orders, and Dynamics. ArXiv: 1408.5805.
  * [11] B. Farb & J. Franks. Groups of homeomorphisms of one-manifolds III: Nilpotent subgroups. Ergod. Th. & Dynam. Sys, 23 (2003), 1467-1484.
  * [12] Y. Guivarc’h. Croissances polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 333-379.
  * [13] E. Jorquera. A universal nilpotent group of $C^{1}$ diffeomorphisms of the interval. Topology and its Applications. 159 (2012), 2115-2126.
  * [14] E. Jorquera, A. Navas & C. Rivas. On the sharp regularity for arbitrary actions of nilpotent groups on the interval: the case of $N_{4}$. Ergod. Th. & Dynam. Sys. 38 (2018), 180-194.
  * [15] S. H. Kim & T. Koberda. Diffeomorphism groups of critical regularity. Invent. Math. 221(2) (2020), 421–501.
  * [16] S. H. Kim & T. Koberda. Structure and regularity of group actions on one-manifolds. Springer Monographs in Mathematics (2021).
  * [17] S. H. Kim ,T. Koberda & C. Rivas. Direct products, overlapping actions and critical regularity. J. Mod. Dyn. 17 (2021), 285-304.
  * [18] N. Kopell. Commuting diffeomorphisms. In Global Analisys. (Berkeley, CA, 1968), Proc. Sympos. Pure Math., Vol XIV, pp 165-184. Amer. Math. Soc. Providence, RI, 1970.
  * [19] K. Mann & M. Wolff. Reconstructing maps from groups. Arxiv:1907.03024 (2019).
  * [20] A. Navas. Groups of Circle Diffeomorphisms. Chicago Lectures in Mathematics. Univ. of Chicago Press (2011).
  * [21] A. Navas. On centralizers of interval diffeomorphisms in critical (intermediate) regularity. J. Anal. Math. 121 (2013), 1-30.
  * [22] A. Navas. A finitely generated, locally indicable group with no faithful action by $C^{1}$ diffeomorphisms of the interval. Geom. & Topol. 14, (2010), 573-584.
  * [23] K. Parkhe. Nilpotent dynamics in dimension one: Structure and smoothness. Ergod. Th. & Dynam. Sys. 36 (2016), 2258-2272.
  * [24] D. Pixton. Nonsmoothable, unstable group actions. Trans. AMS 229 (1977), 259-268.
  * [25] C. Rivas & M. Triestino. One dimensional actions of Higman’s group. Discrete Analysis (2019), 15pp.
  * [26] J. Plante & W. Thurston. Polynomial growth in holonomy groups of foliations.Comment. Math. Helv. 51 (1976), 567-584.
  * [27] W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347-352.
  * [28] T. Tsuboi. Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle. J. Math. Soc. Japan 47 (1995), 1-30.

Maximiliano Escayola

Dpto. de Matemática y C.C., Universidad de Santiago de Chile (USACH)

Alameda Libertador Bernardo O’Higgins 3363, Estación Central, Santiago, Chile

Email<EMAIL_ADDRESS>
Cristóbal Rivas

Dpto de Matemáticas, Universidad de Chile

Las Palmeras 3425, Ñuñoa, Santiago, Chile

Email<EMAIL_ADDRESS>