# A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity
of solvable groups

Tullia Dymarz  Department of Mathematics, University of Wisconsin-Madison, 480
Lincoln Drive, Madison, WI 53706<EMAIL_ADDRESS>,  David Fisher
Department of Mathematics, Rice University, 6100 Main Street Houston, TX 77005
<EMAIL_ADDRESS>and  Xiangdong Xie  Department of Mathematics and
Statistics, Bowling Green State University, Bowling Green, OH 43403
<EMAIL_ADDRESS>
###### Abstract.

In this paper we study uniform quasiconformal groups of Carnot-by-Carnot
groups. We show that they can be conjugated into conformal groups provided the
induced action on the space of distinct pairs is cocompact. Following the
approach of Eskin-Fisher-Whyte [EFW07] these results have applications to
quasi-isometric rigidity of certain families of solvable groups.

###### Key words and phrases:

uniform quasisimilarity groups, quasi-isometric rigidity, nilpotent Lie
groups, solvable Lie groups

###### Contents

  1. 1 Introduction
  2. 2 Outline of main theorem
  3. 3 Preliminary
    1. 3.1 Quasi-actions and uniform quasisimilarity actions
    2. 3.2 Nilpotent Lie algebras and nilpotent Lie groups
    3. 3.3 Carnot algebras and Carnot groups
    4. 3.4 Derivations, semi-direct products, Heintze groups, and SOL-like groups
    5. 3.5 Homogeneous distances on nilpotent Lie groups
    6. 3.6 A fiber Tukia theorem for diagonal Heintze pairs
  4. 4 Quasi-isometric rigidity of solvable groups
    1. 4.1 Quasi-isometric rigidity of lattices in the isometry group of SOL-like groups
    2. 4.2 Quasi-isometric classification of a class of solvable Lie groups
    3. 4.3 Quasi-isometric rigidity of quasi-actions on certain Heintze groups
  5. 5 BiLipschitz maps of diagonal Heintze pairs
  6. 6  Characterization of biLipschitz shear maps
    1. 6.1 Horizontal tautological one form on Carnot groups
    2. 6.2 Structure of Lie algebra when $\alpha$ is not an integer
    3. 6.3 Characterization of biLipschitz shear maps
  7. 7 Eliminating $s_{j}$ for $j<\alpha$
    1. 7.1 The affine action on $E_{j}$
    2. 7.2 Existence of fixed point in $\mathcal{H}_{j}$
    3. 7.3 Eliminating $s_{j}$
  8. 8 Conformal structures in the $V_{\alpha}$ direction
    1. 8.1 Differential of $F_{p,\alpha}$
    2. 8.2 Measurable conformal structure in the $V_{\alpha}$ direction
    3. 8.3 Completing the proof of Theorem 1.2 when $\dim(W)\geq 2$ and $\dim(N/W)\geq 2$
  9. 9 Case $\text{dim}(W)=1$
  10. 10 Case $\text{dim}(N/W)=1$
  11. A Examples of Carnot-by-Carnot groups
  12. B Lattices in SOL-like groups
  13. C Compatible expressions

##  1\. Introduction

Motivation. The motivation for this paper comes from Gromov’s program of
classifying finitely generated groups up to _quasi-isometric equivalence_. A
map $f:X\to Y$ between metric spaces is a quasi-isometry if there are $A\geq
0$ and $L\geq 1$ such that for all $x_{1},x_{2}\in X$

$-A+d(x_{1},x_{2})/L\leq d(f(x_{1}),f(x_{2}))\leq L\ d(x_{1},x_{2})+A$

and for all $y\in Y$ there is $x\in X$ with $d(f(x),y)\leq A$. In [EFW07]
Eskin-Fisher-Whyte announced quasi-isometric rigidty for lattices in
$SOL=\mathbb{R}^{2}\rtimes\mathbb{R}$, the simplest non-nilpotent unimodular
solvable Lie group and in [EF10] they outlined a conjecture and program for
proving a quasi-isometric classification for all non-nilpotent polycyclic
groups (or equivalently all unimodular non-nilpotent solvable Lie groups $G$).
Along with the quasi-isometric classification of nilpotent groups this remains
one of the most important open problems in Gromov’s Program.

The strategy outlined in [EF10] has two main steps:

  1. (1)

Understand the structure self quasi-isometries of the solvable Lie group $G$.

  2. (2)

Use this structure to identify certain groups of self quasi-isometries of $G$
as conjugates of groups of isometries of a possibly different Lie group
$\bar{G}$.

In the case of SOL the second step is proved in a short appendix to a paper by
Farb-Mosher [FM99]. In many other cases this second step requires detailed
quasiconformal analysis on a homogeneous group $N$ resulting in a so called
_Tukia-type_ theorem (see below). Recall that a homogeneous group is a
nilpotent Lie group together with a dilation that induces a homogenous metric
on $N$. Previous work has been in the case when either $N$ is abelian [Dy10]
or in the Carnot case [DyX16] which is the case where the dilation is
completely determined by the algebra of $N$. In this paper we present the
first case of understanding this analysis for non-abelian non-Carnot
homogeneous groups. We call this case the _Carnot-by-Carnot_ case (Definition
1.1). This case has significant new challenges that involve subtle
interactions between the algebra of the homogeneous group (coming from $N$
being non abelian) and the geometry of the homogeneous metric (coming from $N$
being non-Carnot). For this we introduce new analysis techniques that we
expect will provide the foundation to prove the necessary results on more
general homogeneous Lie groups. Once complete this program will establish the
quasi-isometric classification of all _SOL-like_ solvable Lie groups
(Definition 1.4).

Our main theorems are Theorem 1.2 and Theorem 1.5 below. Theorem 1.5 is quasi-
isometric rigidity for our new family of SOL-like groups while Theorem 1.2,
whose proof is the bulk of this paper, provides Step (2) for Theorem 1.5.
Concrete examples of both Carnot-by-Carnot groups and lattices in the
accompanying SOL-like groups can be found in appendices A and B respectively.

Tukia-type Theorems. A Tukia-type theorem asserts that a group of self maps of
a metric space $(X,d)$ that distort distances by a uniformly controlled amount
must actually be, up to conjugation, a lot more rigid. In this paper, our
uniformly controlled maps will be _quasi-similarities_ , i.e. bijections
$f:(X,d)\to(X,d)$ such that

(1) $K_{f}^{-}d(x,y)\leq d(f(x),f(y))\leq K_{f}^{+}d(x,y)\quad\textrm{ for all
}x,y\in X$

where $K_{f}^{+}/K_{f}^{-}\leq K$ and $K$ is a uniform constant over all group
elements. A more common term for a map satisfying Equation (1) is
_biLipschitz_ but then $K_{f}^{-}$ is usually defined to be the inverse of
$K_{f}^{+}$ so uniformity in our sense is harder to describe. The goal
conclusion will be that up to conjugation by a biLipschitz map, the group must
be acting by _similarities_ , that is maps where $K_{f}^{-}=K_{f}^{+}$. Such a
theorem is always true for example when $X=\mathbb{R}$ [FM99] but in general,
additional hypotheses on the size of the group are often needed.

Tukia’s original theorem [T86] was proved for groups of uniform quasiconformal
maps of the $n$-sphere for $n\geq 2$, showing that up to conjugation these
groups act by conformal maps. For $n=2$ this was also proved by Sullivan
[S78]. When $n\geq 3$ the additional hypothesis that was needed was that every
point be a _radial limit point_ (see Definition 8.5) under the action. We will
also need such a hypothesis in our case. Tukia’s theorem was first generalized
in [Ch96] to quasiconformal groups of the real Heisenberg group equipped with
a Carnot-Caratheodory metric. A similar generalization holds for all Carnot
groups endowed with a Carnot-Caratheodory metric (see Theorem 3.6) but in the
conclusion the conformal action may be with respect to a different Carnot-
Caratheodory metric. Other generalizations of Tukia’s theorem for uniform
quasisimilarity groups may be found in [DyX16] where $N=\mathbb{R}^{n}$ with
homogeneous distances that are not the standard Euclidean distances. This
paper contains the first Tukia-type theorem for non-abelian non-Carnot
homogeneous groups.

######  Definition 1.1.

Let $N$ be a simply connected nilpotent Lie group with Lie algebra
$\mathfrak{n}$. Let $D$ be a diagonalizable derivation of $\mathfrak{n}$ with
positive eigenvalues. Let $\mathfrak{w}$ be the Lie sub-algebra generated by
the eigenspace of $D$ associated with the smallest eigenvalue. We say that
$(N,D)$ is a _Carnot-by-Carnot_ group if the following conditions are
satisfied:

  1. (1)

$\mathfrak{w}$ is a proper ideal of $\mathfrak{n}$;

  2. (2)

$\mathfrak{n}/\mathfrak{w}$ is a Carnot algebra;

  3. (3)

$D$ induces a derivation $\bar{D}$ on $\mathfrak{n}/\mathfrak{w}$ that is a
multiple of the standard Carnot derivation on $\mathfrak{n}/\mathfrak{w}$.

We remark that $\mathfrak{w}$ is itself a Carnot algebra, see Lemma 3.3.
Carnot-by-Carnot groups are abundant. The obvious examples are given by a
direct product of two Carnot groups (where the derivation is a product of
distinct multiples of the standard Carnot derivation on the Lie algebras of
the two factors). More interesting examples are provided by central products
and semi-direct products. Using Lie algebra cohomology one can also obtain
Carnot-by-Carnot groups that are not semi-direct products. See Appendix A for
more details.

Borrowing terminology from [LDNG21], we say a distance $d$ on $N$ is _$D$
-homogeneous_ if it is left invariant, induces the manifold topology on $N$
and such that $d(e^{tD}x,e^{tD}y)=e^{t}d(x,y)$ for all $x,y\in N$ and
$t\in\mathbb{R}$, where $\\{e^{tD}|t\in\mathbb{R}\\}$ is the one parameter
group of automorphisms of $N$ generated by the derivation $D$. There are many
$D$-homogeneous distances $d$ on $N$ and it is easy to see that any two
$D$-homogeneous distances are pairwise biLipschitz equivalent. When we talk
about a biLipschitz map $F$ of $N$ we mean that $F$ is biLipschitz with
respect to a $D$-homogeneous distance on $N$. Note, however, that for two
$D$-homogeneous distances $d_{1}$, $d_{2}$ on $N$, a map
$F:(N,d_{1})\to(N,d_{1})$ being a similarity does not imply that
$F:(N,d_{2})\to(N,d_{2})$ is a similarity. A $D$-homogeneous distance $d_{0}$
on $N$ is _maximally symmetric_ if for any $D$-homogeneous distances $d$ on
$N$ there is a biLipschitz automorphism $\phi$ such that
$\phi\text{Sim}(N,d)\phi^{-1}\subset\text{Sim}(N,d_{0})$, where
$\text{Sim}(N,d)$ denotes the group of similarities of $(N,d)$. We remark that
$N$ always admits a maximally symmetric $D$-homogeneous distance $d_{0}$, see
Lemma 2.3, [DFX].

######  Theorem 1.2.

Let $(N,D)$ be a Carnot-by-Carnot group and $\Gamma$ a group with a uniform
quasisimilarity action of $N$ such that almost every point in $N$ is a radial
limit point. When $\dim(\mathfrak{w})=1$, we further assume $\Gamma$ is
locally compact amenable. Then there exists a biLipschitz map $F_{0}$ of $N$
such that $F_{0}\Gamma F_{0}^{-1}\subset\text{Sim}(N,d_{0})$, where $d_{0}$ is
a fixed maximally symmetric $D$-homogeneous distance on $N$.

We remark that a version of Theorem 1.2 for uniform quasiconformal groups is
not needed since by [CP17] when $N$ is Carnot-by-Carnot, each quasiconformal
map of $N\cup\\{\infty\\}$ fixes $\infty$ and a uniform quasiconformal group
of $N\cup\\{\infty\\}$ is the same thing as a uniform quasisimilarity group of
$N$. Theorem 1.2 also holds for Carnot groups but in this case all that is
required is running the proof of Theorem 3.6 with the additional assumption
that all maps in question fix $\infty$ and are biLipschitz. In a general
homogeneous group there are two sources of metric distortion, one from
brackets and one from differing rates of contraction from the derivation $D$.
The Carnot case is exactly the one where these two distortions agree. A major
difficulty in going beyond the Carnot case [DFX] and the abelian case [Dy]
comes when it is difficult to separate the two sources of distortion. The case
of general $(N,D)$ involves more phenomena of this kind not already seen in
the Carnot-by-Carnot case. In the case of products of Carnot groups this is
easier than in the general Carnot-by-Carnot case. In fact we can easily prove
a similar theorem for uniform quasisimilarity groups of $(N,D)$ with
$N=\prod_{i}N_{i}$ where each $N_{i}$ is Carnot with Lie algebra
$\mathfrak{n}_{i}$ and $D|_{\mathfrak{n}_{i}}$ is an increasing multiple of a
Carnot derivation (see Theorem 10.1).

The amenability condition when $\dim(\mathfrak{w})=1$ is needed in the proof
in order to apply Day’s fixed point theorem which states that affine actions
of amenable groups on compact convex subsets of locally convex topological
vector spaces have fixed points. We actually also use Day’s fixed point
theorem in the case $\dim(\mathfrak{w})\geq 2$ but in that case we are able to
deduce the amenability of the acting group. This extra amenability condition
does not affect the application of our theorem to quasi-isometric rigidity
since amenability of discrete groups is preserved by quasi-isometries.
Nevertheless it would be interesting to know if this condition can be removed.

The proof of Theorem 1.2 takes up the bulk of the paper and so we outline it
in more detail in the next section.

Quasi-isometric rigidity of solvable groups. The power of a Tukia-type theorem
in geometric group theory is that it can often be used to prove quasi-
isometric rigidity results. This is done through the notion of quasi-action. A
quasi-action of a group $\Gamma$ on a metric space $X$ is an assignment
$\gamma\mapsto G_{\gamma}$ where $G_{\gamma}$ is a self quasi-isometry of $X$
such that

  1. (1)

$G_{\gamma}$ is an $(L,A)$ quasi-isometry where $L$ and $A$ are uniform over
all $\gamma\in\Gamma$.

  2. (2)

$G_{\gamma\eta}$ and $G_{\gamma}\circ G_{\eta}$ are bounded distance apart in
the sup norm, uniformly over all $\gamma,\eta\in\Gamma$.

  3. (3)

$G_{Id}$ is bounded distance from the identity map on $X$.

A quasi-action is _cobounded_ if there is a bounded set $S\subset X$ such that
for any $x\in X$ there is $\gamma\in\Gamma$ such that $G_{\gamma}(x)\in S$.

The standard example of a cobounded quasi-action arises when $\Gamma$ is a
group with a left invariant metric (for example, a finitely generated group
with a word metric or a Lie group with a left invariant Riemannian metric) and
$\phi:\Gamma\to X$ is a quasi-isometry with coarse inverse $\bar{\phi}$. Then
$\gamma\mapsto\phi\circ L_{\gamma}\circ\bar{\phi}$ defines a cobounded quasi-
action of $\Gamma$ on $X$, where $L_{\gamma}$ is the left translation of
$\Gamma$ by $\gamma$.

In the case that $X$ is Gromov hyperbolic, quasi-isometries of $X$ correspond
to quasiconformal maps (quasisymmetric maps, to be more precise) of the ideal
boundary $\partial X$, and quasi-actions on $X$ correspond to uniform
quasiconformal actions on $\partial X$. Furthermore, with suitable choices of
metrics on $X$ and $\partial X$, isometries of $X$ often correspond to
conformal maps of $\partial X$ or similarities of a one-point complement of
$\partial X$. As a result, a group $\Gamma$ quasi-acting on a Gromov
hyperbolic space $X$ induces a uniform quasiconformal action on $\partial X$;
if a Tukia-type theorem is available, then this uniform quasiconformal action
is conjugate by a quasiconformal map of $\partial X$ to a conformal action of
$\Gamma$ on $\partial X$, which implies the original quasi-action is conjugate
by a quasi-isometry to an isometric action on $X$.

We now apply the outline of the proceeding paragraph in the case when $X$ is a
so called _Heintze group_ : Given a simply connected nilpotent Lie group $N$
with Lie algebra $\mathfrak{n}$ and a derivation $D$ of $\mathfrak{n}$, one
can form the semi-direct product $S=N\rtimes_{D}\mathbb{R}$, where the action
of $\mathbb{R}$ on $N$ is by the automorphisms $e^{tD}$ of $N$ generated by
the derivation $D$. Such a group $S$ is called a Heintze group if the
eigenvalues of $D$ all have positive real parts. For example when $(N,D)$ is
Carnot-by-Carnot the associated extension $S$ is a Heintze group. By [H74],
Heintze groups are Gromov hyperbolic. By applying our Tukia-type theorem we
get the following (a similar statement holds when $N$ is a Carnot group, see
Corollary 1.2 in [DFX]):

######  Theorem 1.3.

Let $(N,D)$ be Carnot-by-Carnot and $S=N\rtimes_{D}\mathbb{R}$ the associated
Heintze group. Then there is a left invariant Riemannian metric $g_{0}$ on $S$
with the following property. If $\Gamma$ is a group that quasi-acts
coboundedly on $S$ (and when $\dim(W)=1$, we further assume $\Gamma$ is
locally compact amenable) then this quasi-action is quasi-conjugate to an
isometric action of $\Gamma$ on $(S,g_{0})$.

More generally, Tukia-type theorems have been used in the proofs of quasi-
isometric rigidity of certain solvable Lie groups that are not themselves
negatively curved but instead are foliated by negatively curved spaces. Most
notably Eskin-Fisher-Whyte’s quasi-isometric rigidity theorem for SOL [EFW07]
uses the fact that SOL is foliated by hyperbolic planes $\mathbb{H}^{2}$ whose
visual boundary can be identified with
$S^{1}\simeq\mathbb{R}\cup\\{\infty\\}$. Recall that
$SOL\simeq\mathbb{R}^{2}\rtimes\mathbb{R}$ where the action of $\mathbb{R}$ on
$\mathbb{R}^{2}$ is given by $e^{tA}$ where $A$ is the diagonal matrix with
$1$ and $-1$ on the diagonal and a left invariant metric can be given by
$ds^{2}=e^{-2t}dx^{2}+e^{2t}dy^{2}+dt^{2}$. There are two foliations by
hyperbolic planes, given by fixing either the $x$ or the $y$ coordinate. As a
consequence of Eskin-Fisher-Whyte’s coarse differentiation theorem on the
structure of quasi-isometries of SOL, a quasi-action on SOL induces two quasi-
actions on the hyperbolic plane (and hence two quasisimilarity actions on
$\mathbb{R}$), one for each foliation.

######  Definition 1.4.

We call a group _SOL-like_ if it has the form $(N_{1}\times
N_{2})\rtimes\mathbb{R}$, where $N_{i}$ is a simply connected nilpotent Lie
group with Lie algebra $\mathfrak{n}_{i}$ and the action of $\mathbb{R}$ on
$N_{1}\times N_{2}$ is given by $e^{tD}$ with $D$ the derivation of
$\mathfrak{n}_{1}\times\mathfrak{n}_{2}$ given by $D:=(D_{1},-D_{2})$ and
$D_{i}$ a derivation of $\mathfrak{n}_{i}$ whose eigenvalues have positive
real part.

The negatively curved spaces foliating $(N_{1}\times N_{2})\rtimes\mathbb{R}$
are simply $N_{i}\rtimes\mathbb{R}$ where the action of $\mathbb{R}$ is
defined by the derivation $D_{i}$. The visual boundary of
$N_{i}\rtimes\mathbb{R}$ is simply $N_{i}\cup\\{\infty\\}$ but in the context
of quasi-isometries of SOL-like groups the point $\infty$ is always preserved.
The structure of quasi-isometries of SOL-like groups follows from Eskin-
Fisher-Whyte’s coarse differentiation techniques [EFW07, EF10] (see also
[Fe22] for a detailed treatment of the “non-unimodular” case in a more general
setting). A consequence of Eskin-Fisher-Whyte’s argument is that any quasi-
action of a group $\Gamma$ on $(N_{1}\times N_{2})\rtimes\mathbb{R}$ induces a
quasi-action of $\Gamma$ on $N_{i}\rtimes\mathbb{R}$ for $i=1,2$. A Tukia-type
theorem (if available) can then be applied to get isometric actions of
$\Gamma$ on $N_{1}\rtimes\mathbb{R}$ and $N_{2}\rtimes\mathbb{R}$. One then
tries to show that the two isometric actions combine to give an isometric
action of $\Gamma$ on $(N_{1}\times N_{2})\rtimes\mathbb{R}$. Using this
strategy and our Tukia-type theorem we get:

######  Theorem 1.5.

Let $S=(N_{1}\times N_{2})\rtimes\mathbb{R}$ be a SOL-like group, where each
$(N_{i},D_{i})$ is either Carnot or Carnot-by-Carnot. In the case
$(N_{i},D_{i})$ is Carnot-by-Carnot with $\dim(\mathfrak{w}_{i})=1$ for at
least one $i$, where $\mathfrak{w}_{i}$ is the Lie sub-algebra of
$\mathfrak{n}_{i}$ generated by the eigenspace of the smallest eigenvalue of
$D_{i}$, we further assume that $\Gamma$ is amenable or that the isometry
group $\text{Isom}(S,g)$ admits a uniform lattice for some left invariant
Riemannian metric $g$ on $S$. There is a left invariant Riemannian metric
$g_{0}$ on $S$ with the following property. Suppose a finitely generated group
$\Gamma$ is quasi-isometric to $S$. Then $\Gamma$ is, up to finite index and
finite kernel, a lattice in $\text{Isom}(S,g_{0})$.

Note that any two finitely generated groups that differ by finite index or
finite kernel are automatically quasi-isometric so it is impossible to drop
these conditions. The assumption that $\text{Isom}(S,g)$ admits a uniform
lattice implies that $\Gamma$ is amenable and so we can apply Theorem 1.2: by
Theorem 3 in [W00], a uniform lattice $\Gamma_{0}$ in $\text{Isom}(S,g)$ is
virtually solvable; on the other hand, $\Gamma$ is quasi-isometric to $S$ and
so is also quasi-isometric to $\Gamma_{0}$; since amenability is a quasi-
isometry invariant among finitely generated groups it follows that $\Gamma$ is
amenable. In Appendix B we give examples of SOL-like groups $S$ that admit
lattices (in this case $\text{Isom}(S,g)$ admit uniform lattices for any left
invariant Riemannian metric $g$). The proof of Theorem 1.5 can be found in
Section 4.

Theorem 1.5 is related to Conjecture 1.2.2(2) in [LDPX22]. Theorem 1.5 also
has consequences for SOL-like groups whose isometry groups do not admit
lattices.

######  Conjecture 1.6.

Let $S$ be a SOL-like group. Suppose $\text{Isom}(S,g)$ does not admit a
uniform lattice for any left invariant Riemannian metric $g$ on $S$. Then $S$
is not quasi-isometric to any finitely generated group.

By Theorem 1.5 this conjecture is true when both $N_{i}$ are either Carnot or
Carnot-by-Carnot with $\dim(\mathfrak{w}_{i})\geq 2$. In case
$\dim(\mathfrak{w}_{i})=1$ we can only conclude that $S$ is not quasi-
isometric to any amenable finitely generated group.

Quasi-isometric classification of solvable Lie groups. An open problem
concerning the large scale geometry of Lie groups is the quasi-isometry
classification of simply connected solvable Lie groups. A simply connected
solvable Lie group $S$ with Lie algebra $\mathfrak{s}$ is of _real type_ (also
called completely solvable, or split-solvable) if all eigenvalues of
$\text{ad}(X)$ are real for all $X\in\mathfrak{s}$. By Theorem 4.21 in
[CKLGO21] for any simply connected solvable Lie group $S$, there is a unique
simply connected solvable Lie group $S_{\mathbb{R}}$ of real type, so called
_real shadow_ of $S$, such that $S$ and $S_{\mathbb{R}}$ can be made isometric
to each other. Recall that two Lie groups $S_{1}$ and $S_{2}$ can be made
isometric to each other if there are left invariant Riemannian metrics $g_{i}$
on $S_{i}$ ($i=1,2$) such that $(S_{1},g_{1})$ and $(S_{2},g_{2})$ are
isometric. Notice that this result also implies two simply connected solvable
Lie groups $S_{1}$ and $S_{2}$ of real type can be made isometric if and only
if they are isomorphic. It follows that to classify simply connected solvable
Lie groups up to quasi-isometry, it suffices to restrict attention to the
class of simply connected solvable Lie groups of real type. Here is a
conjecture by Y. Cornulier.

######  Conjecture 1.7.

([dC18], Conjecture 19.113) Two simply connected solvable Lie groups $S_{1}$
and $S_{2}$ of real type are quasi-isometric if and only if they are
isomorphic.

The following two results confirm the above conjecture for two classes of
solvable Lie groups.

######  Theorem 1.8.

Let $\mathcal{S}$ be the class of SOL-like groups $S=(N_{1}\times
N_{2})\rtimes\mathbb{R}$ where $(N_{i},D_{i})$ ($i=1,2$) is either Carnot or
Carnot-by-Carnot. Then two members $S_{1},S_{2}\in\mathcal{S}$ are quasi-
isometric if and only if they are isomorphic.

In the special case when $S$ is non-unimodular and $(N_{i},D_{i})$ ($i=1,2$)
is Carnot, Theorem 1.8 is the same as Theorem C , [Fe22]. Similarly Theorem
1.9 follows from Pansu’s differentiability theorem in the case when $(N,D)$ is
Carnot.

######  Theorem 1.9.

Let $\mathcal{S}$ be the class of simply connected solvable Lie groups of the
form $S=N\rtimes_{D}\mathbb{R}$, where $(N,D)$ is Carnot or Carnot-by-Carnot.
Then two members $S_{1},S_{2}\in\mathcal{S}$ are quasi-isometric if and only
if they are isomorphic.

As indicated above, a quasi-isometry between $S_{1},S_{2}$ induces a quasi-
action of $S_{1}$ on $S_{2}$. Our Tukia-type theorem and Eskin-Fisher-Whyte’s
coarse differentiation method then imply this quasi-action can be conjugated
into an isometric action of $S_{1}$ on $S_{2}$. From here it is not too
difficulty to show that $S_{1}$ and $S_{2}$ are isomorphic.

Structure of the paper: We collect the preliminaries in Section 3. The proofs
of the quasi-isometric rigidity theorems are covered in Section 4. The proof
of Theorem 1.2 spans from Section 5 to Section 10, see Section 2 for an
outline of this proof. Appendix A provides examples of Carnot-by-Carnot
groups. In Appendix B we review some examples of SOL-like groups that admit
lattices. The content of Appendix C is the proof of Lemma 5.4.

Acknowledgments. T. Dymarz was supported by NSF career Grant 1552234. D.
Fisher was supported by NSF grants DMS-1906107 and DMS-2246556. X. Xie was
supported by Simons Foundation grant #315130. X. Xie would like to thank the
department of mathematics, University of Wisconsin at Madison for financial
support during his visit there in February 2020. We would also like to thank
Dave Witte Morris for useful conversations and thank Tom Ferragut and Gabriel
Pallier for thoughtful comments on an earlier version.

##  2\. Outline of main theorem

In this section we outline the proof of Theorem 1.2.

Let $\Gamma$ be a group acting on a Carnot-by-Carnot group $(N,D)$ by uniform
quasisimilarities with respect to a $D$-homogeneous distance that satisfies
the conditions of Theorem 1.2. We use the notation $W$, $\mathfrak{w}$ and
$\bar{D}$ from Definition 1.1 for a Carnot-by-Carnot group. Without loss of
generality we can assume that $D$ scales by $1$ on the first layer of
$\mathfrak{n}$ (equivalently the first layer of $\mathfrak{w}$) and by
$\alpha$ on the first layer of $\mathfrak{n}/\mathfrak{w}$. For example if
$N\simeq\mathbb{R}^{n}$ then $D$ is a diagonal matrix with diagonal entries
$1$ and $\alpha$. (see Section 3 for details).

A biLipschitz map $f$ of $N$ is a _fiber similarity map_ if there exist
Carnot-Caratheodory metrics $\bar{d}_{CC}$ and $d_{CC}$ on $N/W$ and $W$
respectively such that

  1. (1)

$f$ induces a similarity of $(N/W,\bar{d}_{CC})$,

  2. (2)

$f$ acts by similarities along cosets of $W$, i.e. for each $g\in N$ the map
$L_{f(g)^{-1}}\circ f\circ L_{g}:(W,d_{CC})\to(W,d_{CC})$ is a similarity.

We say $\Gamma$ is a _fiber similarity group_ if there exist Carnot-
Caratheodory metrics $\bar{d}_{CC}$ and $d_{CC}$ on $N/W$ and $W$ respectively
such that every element $\gamma\in\Gamma$ is a fiber similarity map with
respect to $\bar{d}_{CC}$ and $d_{CC}$. When $\dim(N/W)\geq 2,\dim(W)\geq 2$,
Theorem 3.7 implies that $\Gamma$ can be conjugated into a fiber similarity
group.

Individual biLipschitz maps (Section 5). In Section 5 we study the structure
of individual fiber similarity maps of $N$. In general fiber similarity maps
are not similarities of $N$ since the similarity induced on $N/W$ may not be
compatible with the similarities on the cosets of $W$ and there may be a shear
along the cosets of $W$. When $N$ is abelian, one can show that any fiber
similarity map is the composition of a similarity of $N$ and a shear map
$(w,h)\mapsto(w+s(h),h)$ along cosets of $W$ (these are called “almost
similarities” in [Dy10]). The same is true if $N$ is a direct product of $W$
and $H$: up to left translation, any such map has the form
$(w,h)\mapsto(\phi(w)s(h),B(h))$ where $\phi:W\to W$ and $B:H\to H$ are
automorphisms, $(w,h)\mapsto(\phi(w),B(h))$ is an automorphism of $N$ (and a
similarity with respect to some $D$-homogeneous distance) and $s:H\to W$ is
the shear amount.

The general case is more involved. To understand the interaction of $W$ and
$N/W$ we decompose the Lie algebras $\mathfrak{n}$ of $N$ and $\mathfrak{w}$
of $W$ into eigenspaces

$\mathfrak{n}=V_{\lambda_{1}}\oplus\cdots\oplus V_{\lambda_{n}}\textrm{ and
}\mathfrak{w}=W_{1}\oplus\cdots\oplus W_{m}.$

Here $V_{\lambda_{j}}$ is the eigenspace associated to the eigenvalue
$\lambda_{j}$ of $D$ and $W_{i}$ is the eigenspace of $D|_{\mathfrak{w}}$
associated to eigenvalue $i$. Recall that without loss of generality we have
assumed that $\lambda_{1}=1$ so that, since ${\mathfrak{w}}$ is Carnot with
respect to $D|_{\mathfrak{w}}$, its eigenvalues are all positive integers. The
eigenvalue $\lambda_{j}$ will either be an integer or a multiple of $\alpha$
(or both) where $\alpha$ is the smallest eigenvalue of the induced derivation
$\bar{D}$ on the Lie algebra of $N/W$.

If $k$ is not an integral multiple of $\alpha$ then $V_{k}=W_{k}$ otherwise
$k=j\alpha$ and we write

$V_{k}=W_{j\alpha}\oplus H_{j}$

for some complement $H_{j}\subset V_{k}$ of $W_{j\alpha}$ (see Section 3 for
details). Then we have a bijection from $H=H_{1}\oplus\cdots\oplus H_{s}$ to
$\mathfrak{n}/\mathfrak{w}$ so that $H$ is a transversal for $\mathfrak{w}$ in
$\mathfrak{n}$. Note that a fiber similarity map does not necessarily preserve
$H$. In Lemma 5.4 we show that any such map must have a _compatible
expression_. For a rigorous definition of compatible expression see Definition
5.3 but morally speaking it still involves an automorphism $\phi:W\to W$ and a
shear map on $N$ determined by a map $s:N/W\to W$ but now $B:H\to\mathfrak{n}$
is only a linear map from the transversal $H$ into $\mathfrak{n}$ that
satisfies the following compatibility condition with $\phi$: for any
$w\in\mathfrak{w}$ and $h\in H$, $[d\phi(w),Bh]=d\phi[w,h]$. In particular
$w+h\mapsto d\phi(w)+Bh$ does not necessarily define an automorphism of
$\mathfrak{n}$. In addition both $B$ and the shear $s$ depend on the choice of
transversal $H$.

Case when $\alpha$ is not an integer. Note that when $\alpha$ is not an
integer then $V_{\alpha}=H_{1}$ and $\mathfrak{n}$ is a central product (see
Section 6.2). This simplifies many of our arguments. In fact a substantial
portion of our later analysis takes place on $V_{\alpha}=W_{\alpha}\oplus
H_{1}$ when $W_{\alpha}\neq 0$.

Shear maps (Section 6). In Section 6 we characterize maps $s:N/W\to W$ for
which the corresponding shear maps $g\mapsto g\cdot s(gW)$ are biLipschitz
maps of $N$. BiLipschitz shear maps are used to conjugate a uniform
quasisimilarity group to eliminate the $W_{j}$ shear components for $j<\alpha$
in the compatible expression of group elements. As above we work with Lie
algebra coordinates, we identify $W$ with $\mathfrak{w}$ and $N/W$ with
$\mathfrak{n}/\mathfrak{w}$ via the exponential map and write $s$ as
$s:\mathfrak{n}/\mathfrak{w}\rightarrow\mathfrak{w}$. First we show that all
such maps $s$ must have image in $Z(\mathfrak{w})$, the center of
$\mathfrak{w}$. In the abelian case any $\frac{1}{\alpha}$-Hølder map
$\mathfrak{n}/\mathfrak{w}\to Z(\mathfrak{w})$ induces a biLipschitz map of
$N$. When $\alpha$ is not an integer there is also a simple description of
such admissible maps $s$. In general, the description is more complex. To
describe $s:\mathfrak{n}/\mathfrak{w}\to Z(\mathfrak{w})$ we first decompose
$s=(s_{i})$ where $s_{i}:H\to Z_{i}:=W_{i}\cap Z(\mathfrak{w})$. Then we show
that $s$ is admissible as long as each $s_{i}$ lies in a function space
$\mathcal{H}_{i}$ that is recursively defined by the vanishing of certain
integrals along horizontal closed curves. (See Section 6.3 and especially
Propositions 6.4 and 6.7 for details).

Day’s theorem (Section 7). Now we are ready to look at the entire uniform
group $\Gamma$, acting by fiber similarity maps on $N$. The goal in this step
is to show that after conjugation the shear components $s_{j}$ for $j<\alpha$
of each group element must be zero. This is where Day’s theorem comes into
play. Recall that Day’s theorem shows that any amenable group that has an
affine action on a compact convex subset of a locally convex topological
vector space has a fixed point. In previous cases, such as those covered in
[DyX16], Day’s theorem applies directly to the action of $\Gamma$ on the
Banach space of $\frac{1}{\alpha}$-Holder functions. Using the fixed point
obtained, one can use the corresponding biLipschitz shear map to conjugate the
group action to an action where all shear components are trivial. In this
paper we need to show that the fixed point lies in $\mathcal{H}_{j}$, the
subspace of Hølder continuous maps defined above. We do this in order to
construct a conjugating biLipschitz shear map, see the preceding paragraph.

Conformal structure in the $V_{\alpha}$ direction (Section 8). At this point,
the shear components $s_{j}$ for $j<\alpha$ of each group element are zero. In
Section 8 we conjugate one more time so that $s_{\alpha}$ is a homomorphism.
This part requires substantially more work. It involves first applying a
modified version of a foliated Tukia-type argument (as in [Dy10] and [DFX])
that uses a measurable conformal structure defined solely in the $V_{\alpha}$
direction. Some care is needed to define this structure since our maps do not
necessarily preserve $V_{\alpha}$. Once we know that $s_{\alpha}$ is a
homomorphism it is not hard to show that our maps are affine maps of $N$ and
hence similarities with respect to some $D$-homogeneous distance on $N$. This
section is only needed when $\alpha$ is an integer.

Dimension one (Sections 9 and 10). Theorem 3.7 only applies when $\dim(N/W)$
and $\dim(W)$ are both at least two. In the case where $\dim(W)=1$ we are not
able to define a foliated conformal structure along $W$ but instead we use the
action of the group on the space of derivative maps in the $W$ direction. Once
again we require Day’s theorem which we apply to this action. We are able to
construct a conjugating map that conjugates our group action into one by fiber
similarity maps. If instead $\dim(N/W)=1$ then there are two cases. If $N$ is
a direct product we can use Farb-Mosher [FM99] to conjugate the action on
$N/W$. Otherwise we use the algebraic structure to show that the group action
was already by fiber similarity maps.

##  3\. Preliminary

###  3.1. Quasi-actions and uniform quasisimilarity actions

Let $L\geq 1$, $A\geq 0$. A (not necessarily continuous) map $f:X\rightarrow
Y$ between two metric spaces is an $(L,A)$-_quasi-isometry_ if:
(1) $d(x_{1},x_{2})/L-A\leq d(f(x_{1}),f(x_{2}))\leq L\,d(x_{1},x_{2})+A$ for
all $x_{1},x_{2}\in X$;
(2) for any $y\in Y$, there is some $x\in X$ with $d(f(x),y)\leq A$.
An $(L,0)$-quasi-isometry is also called a $L$-biLipschitz map.

One of the main motivations for studying quasi-isometries comes from geometric
group theory. A finitely generated group $\Gamma$ with a symmetric finite
generating set $S$ has an associated left invariant _word metric_ given by
$d_{S}(g,h)=\|g^{-1}h\|_{S}$ where $\|a\|_{S}$ denotes the minimal
$k\in\mathbb{Z}_{+}$ such that $a=s_{1}s_{2}\cdots s_{k}$ with $s_{i}\in S$
for $i=1,\ldots,k$. It is a simple exercise to show that for any two finite
generating sets $S,S^{\prime}$ we have that $(\Gamma,d_{S})$ and
$(\Gamma,d_{S^{\prime}})$ are quasi-isometric.

The definition of a quasi-action can be found in the Introduction.

Let $\Lambda\geq 1$ and $C>0$. A bijection $F:X\rightarrow Y$ between two
metric spaces is called a $(\Lambda,C)$-_quasisimilarity_ if

$\frac{C}{\Lambda}\,d(x,y)\leq d(F(x),F(y))\leq C\,\Lambda\,d(x,y)$

for all $x,y\in X$. When $\Lambda=1$, we say $F$ is a similarity. It is clear
that a map is a quasisimilarity if and only if it is a biLipschitz map. The
point of using the notion of quasisimilarity is that sometimes there is
control on $\Lambda$ but not on $C$. An action
$\Phi:\Gamma\rightarrow\text{Homeo}(X)$ of a group $\Gamma$ on a metric space
$X$ is a _uniform quasisimilarity action_ if there is some $\Lambda\geq 1$
with the following property: for every $\gamma\in\Gamma$ there is a constant
$C_{\gamma}>0$ such that $\Phi(\gamma)$ is a
$(\Lambda,C_{\gamma})$-quasisimilarity.

Let $X$ be a Gromov hyperbolic space, and $\Gamma$ a group quasi-acting on
$X$. If there is a point $\infty\in\partial X$ such that for any
$\gamma\in\Gamma$, the homeomorphism of $\partial X$ induced by $G_{\gamma}$
fixes $\infty$ and is biLipschitz with respect to a fixed visual metric on
$\partial X$, then the quasi-action of $\Gamma$ on $X$ induces a uniform
quasisimilarity action of $\Gamma$ on $(\partial X\backslash\\{\infty\\},d)$
for some parabolic visual metric $d$ based at $\infty$. Conversely, if $X$ is
a visual Gromov hyperbolic space and $\Gamma$ admits a uniform quasisimilarity
action on $(\partial X\backslash\\{\infty\\},d)$ for some parabolic visual
metric $d$ based at $\infty$, then this action extends to a quasi-action of
$\Gamma$ on $X$. More generally, there is a notion of uniform quasi-Mobius
actions, and quasi-actions on visual Gromov hyperbolic spaces correspond to
uniform quasi-Mobius actions on the visual boundary. Since we do not need the
notion of uniform quasi-Mobius actions, we shall not recall the definition
here.

###  3.2. Nilpotent Lie algebras and nilpotent Lie groups

Let $\mathfrak{n}$ be a Lie algebra. The lower central series is defined
recursively as follows: ${\mathfrak{n}}^{(1)}=\mathfrak{n}$,
${\mathfrak{n}}^{(k)}=[\mathfrak{n},\mathfrak{n}^{(k-1)}]$. The Lie algebra
$\mathfrak{n}$ is called nilpotent if ${\mathfrak{n}}^{(t+1)}=0$ for some
$t\geq 1$. The smallest such $t$ is called the nilpotency of $\mathfrak{n}$. A
connected Lie group is nilpotent if and only if its Lie algebra is nilpotent.

Let $N$ be a simply connected nilpotent Lie group with Lie algebra
$\mathfrak{n}$. Then the exponential map $\text{exp}:{\mathfrak{n}}\rightarrow
N$ is a diffeomorphism. Under this identification the Lebesgue measure on
$\mathfrak{n}$ is a Haar measure on $N$ ([CG90], page 19). One can pull back
the group operation from $N$ to get a group structure on $\mathfrak{n}$. This
group structure on $\mathfrak{n}$ can be described by the Baker-Campbell-
Hausdorff formula (BCH formula in short), which expresses the product $X*Y$
($X,Y\in{\mathfrak{n}}$) in terms of the iterated Lie brackets of $X$ and $Y$.
The group operation in $N$ will be denoted by $\cdot$. The pull-back group
operation $*$ on $\mathfrak{n}$ is defined as follows. For
$X,Y\in\mathfrak{n}$, define

$X*Y=\text{exp}^{-1}(\text{exp}X\cdot\text{exp}Y).$

Then the BCH formula ([CG90], page 11) says

$\displaystyle X*Y$
$\displaystyle=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+\cdots.$

We also recall the following formula ([Hel78], pages 127, 128):

(2) $Y*X*(-Y)=\sum_{k=0}^{\infty}\frac{1}{k!}({\rm
ad}\,Y)^{k}(X)=X+[Y,X]+\cdots+\frac{1}{k!}({\rm ad}\,Y)^{k}(X)+\cdots.$

Recall, if $\phi:G_{1}\to G_{2}$ is a Lie group homomorphism and
$d\phi:\mathfrak{g}_{1}\to\mathfrak{g}_{2}$ denotes the associated Lie algebra
homomorphism then $\phi\circ\text{exp}=\text{exp}\circ d\phi$.

Standing Assumption. In this paper (particularly in Sections 6–8 and Appendix
C) we shall often identify a simply connected nilpotent Lie group $N$ with its
Lie algebra $\mathfrak{n}$ via the exponential map. With this identification,
the inverse (with respect to $*$) of an element $X\in\mathfrak{n}$ is $-X$:
$X^{-1}=-X$. For this reason we denote by $0$ the identity element of a Lie
group.

The following statement is known (see for example [P11a], p.980). We include
it here for completeness.

######  Lemma 3.1.

Let $\phi,\tilde{\phi}:N_{1}\rightarrow N_{2}$ be Lie group homomorphisms
between two simply connected nilpotent Lie groups. Let $N_{2}$ be equipped
with a left invariant metric $d$ such that $d$ induces the manifold topology
and closed balls are compact. Suppose $C:=\sup_{x\in
N_{1}}\\{d(\phi(x),\tilde{\phi}(x))\\}<\infty$. Then $\phi=\tilde{\phi}$.

###### Proof.

It suffices to show the corresponding Lie algebra homomorphisms agree. Suppose
the contrary that there is some $X\in\mathfrak{n}_{1}$ such that
$d\phi(X)\not=d\tilde{\phi}(X)$. Let $\\{\mathfrak{n}_{2}^{(j)}\\}_{j}$ be the
lower central series of $\mathfrak{n}_{2}$ and
$\pi_{j}:\mathfrak{n}_{2}\rightarrow\mathfrak{n}_{2}/{\mathfrak{n}_{2}^{(j)}}$
the quotient homomorphism. Let $k\geq 2$ be the integer satisfying
$d\phi(X)-d\tilde{\phi}(X)\in\mathfrak{n}_{2}^{(k-1)}\backslash\mathfrak{n}_{2}^{(k)}$.
Set $Y=\pi_{k}(d\phi(X))$, $\tilde{Y}=\pi_{k}(d\tilde{\phi}(X))$ and
$\mathfrak{g}=\mathfrak{n}_{2}/{\mathfrak{n}_{2}^{(k)}}$. Then $Y=\tilde{Y}+Z$
for some $Z\not=0$ in the center of $\mathfrak{g}$. The assumption implies
$\bar{d}(tY,t\tilde{Y})\leq C$ for all $t\in\mathbb{R}$, where $\bar{d}$ is
the distance on $\mathfrak{g}$ induced by $d$:
$\bar{d}(\bar{a},\bar{b})=\inf\\{d(a,b)|a\in\pi_{k}^{-1}(\bar{a}),b\in\pi_{k}^{-1}(\bar{b})\\}$.
Here we identified $N_{2}$ with $\mathfrak{n}_{2}$ and so $d$ is a distance on
$\mathfrak{n}_{2}$. On the other hand, the BCH formula gives
$(-t\tilde{Y})*(tY)=tZ$ and so
$C\geq\bar{d}(tY,t\tilde{Y})=\bar{d}(0,(-t\tilde{Y})*(tY))=\bar{d}(0,tZ)$.
Pick $z\in\pi^{-1}_{k}(Z)$. Then there is some
$a_{t}\in{\mathfrak{n}_{2}^{(k)}}$ satisfying $d(a_{t},tz)\leq C$. We can
write $tz=a_{t}*b_{t}$ for some $b_{t}\in
B:=\\{x\in\mathfrak{n}_{2}|d(0,x)\leq C\\}$. Since ${\mathfrak{n}_{2}^{(k)}}$
is an ideal of $\mathfrak{n}_{2}$, by the BCH formula we can write
$a_{t}*b_{t}=c_{t}+b_{t}$ for some $c_{t}\in{\mathfrak{n}_{2}^{(k)}}$. We have
$tz=c_{t}+b_{t}$ and so $z=\frac{c_{t}}{t}+\frac{b_{t}}{t}$. Since the ball
$B$ is compact, we have $\frac{b_{t}}{t}\to 0$ as $t\to\infty$. This implies
$\frac{c_{t}}{t}\to z$ as $t\to\infty$. As
$\frac{c_{t}}{t}\in{\mathfrak{n}_{2}^{(k)}}$ and ${\mathfrak{n}_{2}^{(k)}}$ is
closed in $\mathfrak{n}_{2}$, we conclude that $z\in{\mathfrak{n}_{2}^{(k)}}$,
contradicting the fact that $Z\not=0$.

∎

###  3.3. Carnot algebras and Carnot groups

Carnot algebras form a particular class of nilpotent Lie algebras. A Carnot
Lie algebra is a finite dimensional Lie algebra $\mathfrak{n}$ together with a
direct sum decomposition $\mathfrak{n}=V_{1}\oplus V_{2}\oplus\cdots\oplus
V_{r}$ of non-trivial vector subspaces such that $[V_{1},V_{i}]=V_{i+1}$ for
all $1\leq i\leq r$, where we set $V_{r+1}=\\{0\\}$. For every Carnot algebra
$\mathfrak{n}=V_{1}\oplus V_{2}\oplus\cdots\oplus V_{r}$, the _Carnot
derivation_ $D:\mathfrak{n}\rightarrow\mathfrak{n}$ is given by $D(x)=ix$ for
$x\in V_{i}$. The automorphisms of $\mathfrak{n}$ generated by the Carnot
derivation are called _Carnot dilations_ and are given by
$\delta_{t}:\mathfrak{n}\rightarrow\mathfrak{n}$, $t\in(0,+\infty)$, where
$\delta_{t}(x)=t^{i}x$ for $x\in V_{i}$. Let $\mathfrak{n}=V_{1}\oplus
V_{2}\oplus\cdots\oplus V_{r}$ and $\mathfrak{n}^{\prime}=V^{\prime}_{1}\oplus
V^{\prime}_{2}\oplus\cdots\oplus V^{\prime}_{s}$ be two Carnot algebras. A Lie
algebra homomorphism $\phi:\mathfrak{n}\rightarrow\mathfrak{n}^{\prime}$ is
graded if $\phi(V_{i})\subset V^{\prime}_{i}$ for all $1\leq i\leq r$.

A simply connected nilpotent Lie group is a Carnot group if its Lie algebra is
a Carnot algebra. Let $N$ be a Carnot group with Lie algebra
$\mathfrak{n}=V_{1}\oplus\cdots\oplus V_{r}$. The subspace $V_{1}$ defines a
left invariant distribution $HN\subset TN$ on $N$. An absolutely continuous
curve $\gamma$ in $N$ whose velocity vector $\gamma^{\prime}(t)$ is contained
in $H_{\gamma(t)}N$ for a.e. $t$ is called a horizontal curve. By Chow’s
theorem ([BR, Theorem 2.4]), any two points of $N$ can be connected by
horizontal curves. Fix a left invariant inner product on $HN$. Let $p,q\in N$,
the _Carnot-Caratheodory metric_ $d_{CC}(p,q)$ between them is defined as the
infimum of length of horizontal curves that join $p$ and $q$. Since the inner
product on $HN$ is left invariant, the Carnot metric on $N$ is also left
invariant. Different choices of inner product on $HN$ result in Carnot metrics
that are biLipschitz equivalent. The Hausdorff dimension of $N$ with respect
to a Carnot metric is given by $\sum_{i=1}^{r}i\cdot\dim(V_{i})$.

######  Definition 3.2.

Let $G$ and $G^{\prime}$ be two Carnot groups endowed with Carnot-Caratheodory
metrics, and $U\subset G$, $U^{\prime}\subset G^{\prime}$ open subsets. A map
$F:U\rightarrow U^{\prime}$ is Pansu differentiable at $x\in U$ if there
exists a graded homomorphism $L:G\rightarrow G^{\prime}$ such that

$\lim_{y\rightarrow x}\frac{d(F(x)^{-1}*F(y),\,L(x^{-1}*y))}{d(x,y)}=0.$

In this case, the graded homomorphism $L:G\rightarrow G^{\prime}$ is called
the Pansu differential of $F$ at $x$, and is denoted by $DF(x)$.

Pansu [P89] showed that a Lipschitz map between Carnot groups is Pansu
differentiable a.e. and if the map is biLipschitz then the Pansu differential
is a graded isomorphism.

###  3.4. Derivations, semi-direct products, Heintze groups, and SOL-like
groups

Let $\mathfrak{n}$ be a Lie algebra. A linear map
$D:\mathfrak{n}\rightarrow\mathfrak{n}$ is called a derivation if
$D[X,Y]=[DX,Y]+[X,DY]$ for all $X,Y\in\mathfrak{n}$. Let
$\text{der}(\mathfrak{n})$ be the set of derivations of $\mathfrak{n}$. It is
clearly a vector space. It becomes a Lie algebra with the usual bracket for
endomorphisms: for $D_{1},D_{2}\in\text{der}(\mathfrak{n})$, define
$[D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}$.

Let $\mathfrak{w}$ and $\mathfrak{h}$ be two Lie algebras, and
$\phi:\mathfrak{h}\rightarrow\text{der}(\mathfrak{w})$ a Lie algebra
homomorphism. The semi-direct product $\mathfrak{w}\rtimes_{\phi}\mathfrak{h}$
is the vector space $\mathfrak{w}\oplus\mathfrak{h}$ with the Lie bracket
given by:

$[(W_{1},H_{1}),(W_{2},H_{2})]=([W_{1},W_{2}]+\phi(H_{1})(W_{2})-\phi(H_{2})(W_{1}),[H_{1},H_{2}]).$

Let $\mathfrak{n}$ be a Lie algebra and $D\in\text{der}(\mathfrak{n})$. Then
we can define a homomorphism
$\phi:\mathbb{R}\rightarrow\text{der}(\mathfrak{n})$ by $\phi(t)=tD$. We
denote
$\mathfrak{n}\rtimes_{D}\mathbb{R}:=\mathfrak{n}\rtimes_{\phi}\mathbb{R}$.
When $\mathfrak{n}$ is a nilpotent Lie algebra and the eigenvalues of
$D\in\text{der}(\mathfrak{n})$ have positive real parts, we call
$\mathfrak{n}\rtimes_{D}\mathbb{R}$ a Heintze algebra. In this case, the
simply connected solvable Lie group with Lie algebra
$\mathfrak{n}\rtimes_{D}\mathbb{R}$ is called a Heintze group, and the pair
$(\mathfrak{n},D)$ (and also $(N,D)$) is called a Heintze pair. A Heintze
algebra (or Heintze group, or the pair $(\mathfrak{n},D)$) is called purely
real if all the eigenvalues of $D$ are real numbers. By [Ale75], every Heintze
group is biLipschitz to a purely real Heintze group. Since we are only
interested in quai-isometries of Heintze groups in this paper, we only need to
consider purely real Heintze groups.

A purely real Heintze pair $(\mathfrak{n},D)$ is called diagonal if $D$ is
diagonalizable.

######  Lemma 3.3.

Suppose $(\mathfrak{n},D)$ is a diagonal Heintze pair and
$0<\lambda_{1}<\cdots<\lambda_{r}$ are the eigenvalues of $D$. Denote by
$V_{\lambda_{i}}$ the eigenspace associated to $\lambda_{i}$. Then
(1) $\mathfrak{n}=\oplus_{i=1}^{r}V_{\lambda_{i}}$ and
$[V_{\lambda_{i}},V_{\lambda_{j}}]\subset V_{\lambda_{i}+\lambda_{j}}$;
(2) Let $\mathfrak{w}$ be the Lie sub-algebra of $\mathfrak{n}$ generated by
$V_{\lambda_{1}}$. Then $\mathfrak{w}$ is a Carnot algebra and
$D(\mathfrak{w})=\mathfrak{w}$. If $\mathfrak{w}$ is an ideal of
$\mathfrak{n}$, then $D$ projects to a derivation $\bar{D}$ of
$\bar{\mathfrak{n}}:=\mathfrak{n}/\mathfrak{w}$, which is diagonalizable with
positive eigenvalues. Furthermore, the smallest eigenvalue of $\bar{D}$ is
strictly larger than $\lambda_{1}$.

###### Proof.

(1) $\mathfrak{n}=\oplus_{i=1}^{r}V_{\lambda_{i}}$ follows from linear algebra
and $[V_{\lambda_{i}},V_{\lambda_{j}}]\subset V_{\lambda_{i}+\lambda_{j}}$
follows from the definition of derivation.

(2) Set $W_{1}=V_{\lambda_{1}}$ and $W_{j}=[W_{1},W_{j-1}]$ for $j\geq 2$. By
(1) we have $W_{j}\subset V_{j\lambda_{1}}$. It follows that
$\mathfrak{w}=\oplus_{j}W_{j}$ is a Carnot grading and
$D(\mathfrak{w})=\mathfrak{w}$. Now assume $\mathfrak{w}$ is an ideal. As
$D(\mathfrak{w})=\mathfrak{w}$, $D$ induces a derivation
$\bar{D}:\bar{\mathfrak{n}}\rightarrow\bar{\mathfrak{n}}$.

Since $\mathfrak{w}$ is graded, that is,
$\mathfrak{w}=\oplus_{i}(\mathfrak{w}\cap V_{\lambda_{i}})$, we have
$\bar{\mathfrak{n}}=\oplus_{i}(V_{\lambda_{i}}/(\mathfrak{w}\cap
V_{\lambda_{i}})$. As $D|_{V_{\lambda_{i}}}$ is multiplication by
$\lambda_{i}$, $\bar{D}$ is also multiplication by $\lambda_{i}$ when
restricted to $V_{\lambda_{i}}/(\mathfrak{w}\cap V_{\lambda_{i}})$. It follows
that $\bar{D}$ is also diagonalizable and its set of eigenvalues is a subset
of the set of eigenvalues of $D$. Since $W_{1}=V_{\lambda_{1}}$ we have
$V_{\lambda_{1}}/(\mathfrak{w}\cap V_{\lambda_{1}})=0$ and so $\lambda_{1}$ is
not an eigenvalue of $\bar{D}$. It follows that the smallest eigenvalue of
$\bar{D}$ is $>\lambda_{1}$.

∎

A Heintze pair $(\mathfrak{n},D)$ (Heintze algebra, Heintze group) is of
Carnot type if it is purely real with $D$ diagonal such that
$\mathfrak{w}=\mathfrak{n}$, where $\mathfrak{w}$ is as in Lemma 3.3. A
Heintze pair is of non-Carnot type otherwise.

Let $(\mathfrak{n}_{i},D_{i})$ ($i=1,2$) be a Heintze pair. Let $D$ be the
derivation of $\mathfrak{n}_{1}\times\mathfrak{n}_{2}$ given by
$D=(D_{1},-D_{2})$. We call
$(\mathfrak{n}_{1}\times\mathfrak{n}_{2})\rtimes_{D}\mathbb{R}$ a SOL-like
algebra and the simply connected solvable Lie group with Lie algebra
$(\mathfrak{n}_{1}\times\mathfrak{n}_{2})\rtimes_{D}\mathbb{R}$ a SOL-like
group.

For any Lie sub-algebra $\mathfrak{h}$ of a Lie algebra $\mathfrak{n}$, the
normalizer of $\mathfrak{h}$ in $\mathfrak{n}$ is defined by:

$N(\mathfrak{h})=\\{x\in\mathfrak{n}:[x,\mathfrak{h}]\subset\mathfrak{h}\\}.$

######  Lemma 3.4.

If a derivation $D$ of a Lie algebra $\mathfrak{n}$ satisfies
$D(\mathfrak{h})\subset\mathfrak{h}$, then $D(N(\mathfrak{h}))\subset
N(\mathfrak{h})$.

###### Proof.

Let $X\in N(\mathfrak{h})$ and $Y\in\mathfrak{h}$ be arbitrary. Then

$[D(X),Y]=D[X,Y]-[X,D(Y)]\in\mathfrak{h}$

implying $D(X)\in N(\mathfrak{h})$.

∎

###  3.5. Homogeneous distances on nilpotent Lie groups

Let $(N,D)$ be a diagonal Heintze pair. A distance $d$ on $N$ is called
$D$-homogeneous if it is left invariant, induces the manifold topology on $N$
and such that $d(e^{tD}x,e^{tD}y)=e^{t}d(x,y)$ for all $x,y\in N$ and
$t\in\mathbb{R}$, where $\\{e^{tD}|t\in\mathbb{R}\\}$ denotes the
automorphisms of $N$ generated by the derivation $D$. Let
$\mathfrak{n}=\oplus_{j}V_{\lambda_{j}}$ be the decomposition of
$\mathfrak{n}$ into the direct sum of eigenspaces of $D$. An inner product
$\left<\cdot,\cdot\right>$ on $\mathfrak{n}$ is called a $D$-inner product if
the eigenspaces corresponding to distinct eigenvalues are perpendicular with
respect to $\left<\cdot,\cdot\right>$. By the construction in Theorem 2 of
[HSi90], given any $D$-inner product $\left<\cdot,\cdot\right>$ on
$\mathfrak{n}$, there is a $D$-homogeneous distance $d$ on $N$ such that
$d(0,x)=\langle x,x\rangle^{\frac{1}{2\lambda_{j}}}$ for $x\in
V_{\lambda_{j}}$. During the course of proof of Theorem 1.2, we will modify
the $D$-inner products several times and so also the corresponding
$D$-homogeneous distances.

It is easy to see that any two $D$-homogeneous distances on $N$ are
biLipschitz equivalent. We will always equip $N$ with a $D$-homogeneous
distance. Hence it makes sense to speak of a biLipschitz map of $N$ without
specifying the $D$-homogeneous distance.

For computational purposes, we also define a function $\rho$ that is
biLipschitz equivalent to a $D$-homogeneous distance $d$. For any $D$-inner
product $\left<\cdot,\cdot\right>$ on $\mathfrak{n}$ define a “norm” on
$\mathfrak{n}$ by

$||v||=\sum_{i}|v_{i}|^{\frac{1}{\lambda_{i}}},$

where $v=\sum_{i}v_{i}$ with $v_{i}\in V_{\lambda_{i}}$. Then define $\rho$ by
$\rho(x,y)=||x^{-1}*y||$. We identify $\mathfrak{n}$ and $N$. Clearly $\rho$
is left invariant, induces the manifold topology and satisfies
$\rho(e^{tD}x,e^{tD}y)=e^{t}\rho(x,y)$ for all $x,y\in\mathfrak{n}$ and
$t\in\mathbb{R}$. It follows that for any $D$-homogeneous distance $d$ on
$\mathfrak{n}$, there is a constant $L\geq 1$ such that
$d(x,y)/L\leq\rho(x,y)\leq L\cdot d(x,y)$ for all $x,y\in\mathfrak{n}$. The
explicit formula for $\rho$ will make the calculations much easier. We will
frequently use $\rho$ rather than $d$ for estimates.

######  Lemma 3.5.

Let $\phi$ be an automorphism of $N$. Then $\phi$ is biLipschitz if and only
if $d\phi$ is “layer preserving”; that is,
$d\phi(V_{\lambda_{j}})=V_{\lambda_{j}}$ for each $j$.

###### Proof.

First suppose $\phi$ is biLipschitz. Let $0\not=v\in V_{\lambda_{j}}$ and
write $d\phi(v)=\sum_{i}x_{i}$ with $x_{i}\in V_{\lambda_{i}}$. Then
$d\phi(tv)=\sum_{i}{tx_{i}}$. We have

$\rho(0,tv)=|v|^{\frac{1}{\lambda_{j}}}|t|^{\frac{1}{\lambda_{j}}}$

and

$\rho(0,d\phi(tv))=\sum_{i}|x_{i}|^{\frac{1}{\lambda_{i}}}|t|^{\frac{1}{\lambda_{i}}}.$

The biLipschitz condition implies $x_{i}=0$ when $i\not=j$ by letting
$t\rightarrow\infty$ or $t\rightarrow 0$.

Conversely assume $d\phi$ is layer preserving. Then there is some constant
$C\geq 1$ such that

(3) $|v|/C\leq|d\phi(v)|\leq C|v|$

for all $v\in V_{\lambda_{j}}$, $\forall j$. Now let $v\in\mathfrak{n}$. Write
$v=\sum_{j}v_{j}$ with $v_{j}\in V_{\lambda_{j}}$. Then
$d\phi(v)=\sum_{j}d\phi(v_{j})$. We have
$\rho(0,v)=\sum_{j}|v_{j}|^{\frac{1}{\lambda_{j}}}$ and
$\rho(0,d\phi(v))=\sum_{j}|d\phi(v_{j})|^{\frac{1}{\lambda_{j}}}$. Now the
claim follows from (3).

∎

An automorphism $\phi$ of $N$ is called graded if it satisfies the condition
in Lemma 3.5.

Let $G$ be a connected Lie group with a left invariant distance $d$ that
induces the manifold topology, and $H$ a closed normal subgroup of $G$. We
define a distance on $G/H$ by
$\bar{d}(xH,yH)=\inf\\{d(xh_{1},yh_{2})|h_{1},h_{2}\in H\\}$. Then $\bar{d}$
is a left invariant distance on $G/H$ that induces the manifold topology and
the quotient map $(G,d)\to(G/H,\bar{d})$ is $1$-Lipschitz. Since $H$ is
normal, we have $\bar{d}(xH,yH)=d(xh_{1},yH)=d(yh_{2},xH)=d_{H}(xH,yH)$ for
any $h_{1},h_{2}\in H$, where $d_{H}$ denotes the Hausdorff distance. If $F$
is a biLipschitz map of $(G,d)$ that permutes the cosets of $H$, then $F$
induces a biLipschitz map $\bar{F}:(G/H,\bar{d})\to(G/H,\bar{d})$ with the
same biLipschitz constant as $F$.

Let $(\mathfrak{n},D)$ be a diagonal Heintze pair and $d$ a $D$-homogeneous
distance on $N$. Assume $\mathfrak{w}$ is an ideal of $\mathfrak{n}$ such that
$D(\mathfrak{w})\subset\mathfrak{w}$. Then $D$ induces a derivation $\bar{D}$
of $\mathfrak{n}/\mathfrak{w}$ and $({\mathfrak{n}/\mathfrak{w}},\bar{D})$ is
a diagonal Heintze pair. In this case, the distance $\bar{d}$ on $N/W$ induced
by $d$ is a $\bar{D}$-homogeneous distance, where $W$ is the Lie subgroup of
$N$ with Lie algebra $\mathfrak{w}$.

###  3.6. A fiber Tukia theorem for diagonal Heintze pairs

Here we recall the Tukia-type theorem for Carnot groups and a fiber version of
Tukia theorem for diagonal Heintze pairs, see [DFX] for more details.

######  Theorem 3.6.

(Theorem 1.1, [DFX]) Let $N$ be a Carnot group and $\hat{N}=N\cup\\{\infty\\}$
the one-point compactification of $N$. There is a left invariant Carnot-
Caratheodory metric $d_{0}$ on $N$ with the following property. Let $G$ be a
uniform quasiconformal group of $\hat{N}$. If the action of $G$ on the space
of distinct triples of $\hat{N}$ is co-compact, then there is some
quasiconformal map $f:\hat{N}\rightarrow\hat{N}$ such that $fGf^{-1}$ consists
of conformal maps with respect to $d_{0}$.

The metric $d_{0}$ has the largest conformal group in the sense that the
conformal group of any left invariant Carnot-Caratheodory metric is conjugated
into the conformal group of $d_{0}$. In general it is not possible to
conjugate a uniform quasiconformal group into the conformal group of an
arbitrary left invariant Carnot-Caratheodory metric, see Section 6, [DFX] for
an example.

Next let $(\mathfrak{n},D)$ be a diagonal Heintze pair. Then there is a
sequence of $D$-invariant Lie sub-algebras
$\\{0\\}=\mathfrak{n}_{0}\subset\mathfrak{n}_{1}\subset\cdots\subset\mathfrak{n}_{s}=\mathfrak{n}$
with the following properties: each $\mathfrak{n}_{i-1}$ is an ideal of
$\mathfrak{n}_{i}$ with the quotient $\mathfrak{n}_{i}/{\mathfrak{n}_{i-1}}$ a
Carnot Lie algebra; $D$ induces a derivation
$\bar{D}:\mathfrak{n}_{i}/{\mathfrak{n}_{i-1}}\rightarrow\mathfrak{n}_{i}/{\mathfrak{n}_{i-1}}$
which is a multiple of the Carnot derivation of
$\mathfrak{n}_{i}/{\mathfrak{n}_{i-1}}$. Let $N_{i}$ be the connected Lie
subgroup of $N$ with Lie algebra $\mathfrak{n}_{i}$. Then $N/{N_{i}}$ is a
homogeneous manifold and the natural map $\pi_{i}:N/{N_{i-1}}\rightarrow
N/{N_{i}}$ is a fiber bundle with fiber the Carnot group $N_{i}/{N_{i-1}}$. We
call the sequence of subgroups $0=N_{0}<N_{1}<\cdots<N_{s}=N$ the preserved
subgroup sequence.

Let $d$ be a $D$-homogeneous distance on $N$. In general $d$ does not induce
any metric on the homogeneous space $N/{N_{i}}$ when $N_{i}$ is not normal in
$N$. Nonetheless, it induces a metric on the fibers $N_{i}/{N_{i-1}}$ of
$\pi_{i}:N/{N_{i-1}}\rightarrow N/{N_{i}}$. Furthermore, every biLipschitz map
$F$ of $N$ permutes the cosets of $N_{i}$ for each $i$. Hence $F$ induces a
map $F_{i}:N/{N_{i}}\rightarrow N/{N_{i}}$ and a bundle map of
$\pi_{i}:N/{N_{i-1}}\rightarrow N/{N_{i}}$. The restriction of $F_{i}$ to the
fibers of $\pi_{i}$ are biLipschitz maps of the Carnot group $N_{i}/{N_{i-1}}$
in the following sense. For each $p\in N$, let $F_{p}=L_{F(p)^{-1}}\circ
F\circ L_{p}$, where $L_{x}$ denotes the left translation of $N$ by $x$.
Notice that the map $(F_{p})_{i-1}:N/{N_{i-1}}\rightarrow N/{N_{i-1}}$
satisfies $(F_{p})_{i-1}(N_{i}/{N_{i-1}})=N_{i}/{N_{i-1}}$. The statement
above simply means
$(F_{p})_{i-1}|_{N_{i}/{N_{i-1}}}:N_{i}/{N_{i-1}}\rightarrow N_{i}/{N_{i-1}}$
is biLipschitz with respect to any left invariant Carnot-Caratheodory metric
on $N_{i}/{N_{i-1}}$.

######  Theorem 3.7.

(Theorem 1.3, [DFX]) Let $(N,D)$ be a diagonal Heintze pair and $\Gamma$ be a
uniform quasisimilarity group of $N$ that acts cocompactly on the space of
distinct pairs of $N$ (or equivalently $\Gamma$ a group that quasi-acts
coboundedly on $S=N\rtimes_{D}\mathbb{R}$). Let $I=\\{i|1\leq i\leq
s,\dim(N_{i}/{N_{i-1}})\geq 2\\}$. Then there exists a biLipschitz map
$F_{0}:N\to N$ and a left invariant Carnot-Caratheodory metric $d_{i}$ on
$N_{i}/{N_{i-1}}$ for each $i\in I$ such that for each $p\in N$ and each $g\in
F_{0}\Gamma F^{-1}_{0}$, the map
$(g_{p})_{i-1}|_{N_{i}/{N_{i-1}}}:(N_{i}/{N_{i-1}},d_{i})\rightarrow(N_{i}/{N_{i-1}},d_{i})$
is a similarity.

##  4\. Quasi-isometric rigidity of solvable groups

In this section we combine Tukia-type theorems with the coarse differentiation
method of Eskin-Fisher-Whyte [EFW12], [EFW13] to establish quasi-isometric
rigidity results for solvable groups.

###  4.1. Quasi-isometric rigidity of lattices in the isometry group of SOL-
like groups

One of the main applications of Tukia-type theorems for nilpotent groups is
quasi-isometric rigidity of lattices in the isometry group of SOL-like groups.
Recall a SOL-like group has the form $S=(N_{1}\times N_{2})\rtimes\mathbb{R}$
and admits two foliations by Heintze groups $N_{1}\rtimes_{D_{1}}\mathbb{R}$
and $N_{2}\rtimes_{D_{2}}\mathbb{R}$.

There is a two step outline for proving the quasi-isometric rigidity for SOL-
like groups:

  1. (1)

Show that up to composition with an isometry any self quasi-isometry of a SOL-
like group is bounded distance from a product map

$(x,y,t)\to(f(x),f(y),t)\textrm{ where }x\in N_{1},\ y\in N_{2},\
t\in\mathbb{R}.$

  2. (2)

Prove a Tukia-type theorem for groups acting by quasi-similarities on $N_{i},\
i=1,2$.

This strategy comes from Eskin-Fisher-Whyte’s seminal work on the quasi-
isometric rigidity of SOL itself. In their work [EFW07, EFW12, EFW13] they
focus primarily on Step (1) and develop a new “coarse differentiation”
technique for understanding quasi-isometries of SOL-like groups. For clarity
their papers focus primarily on the case of the three dimensional SOL and its
discrete cousins the Diestel-Leader graphs. Despite this, their results follow
similarly for all SOL-like groups as already pointed out in [EF10, Section
4.4]. Indeed, in her work [P11b, P11c] on abelian-by-abelian higher rank SOL-
like groups (i.e groups of the form $\mathbb{R}^{n}\rtimes\mathbb{R}^{m}$ with
appropriate conditions on the action of $\mathbb{R}^{m}$) Peng already points
out that Eskin-Fisher-Whyte’s arguments work more generally in the case of
SOL-like groups where $N_{i}$ is abelian. Recently Ferragut in his thesis
[Fe22] has endeavored to find the most general framework for which Eskin-
Fisher-Whyte’s work applies. In doing so he has extended the work of [EFW12]
to the class of _horo-pointed metric measure spaces_ and is looking to do the
same for [EFW13].

We will briefly review in the SOL case how combining steps (1) and (2) gives
quasi-isometric rigidity for lattices in SOL.

Quasi-isometric rigidity outline for SOL. Let $\Gamma$ be a finitely generated
group quasi-isometric to SOL (hence to any lattice in SOL). Then there is an
induced quasi-action of $\Gamma$ on SOL by $L\geq 1,\ C\geq 0$ quasi-
isometries where $L,C$ are uniform over all $\gamma\in\Gamma$. Up to replacing
$\Gamma$ with a subgroup of index two we can apply Eskin-Fisher-Whyte (i.e.
Step (1) ) to get that all $\gamma\in\Gamma$ act by maps of the form
$\gamma(x,y,t)=(f_{\gamma}(x),g_{\gamma}(y),t+c_{\gamma})$. This induces two
uniform quasisimilarity actions on $\mathbb{R}$ (via $f_{\gamma}$ and
$g_{\gamma}$ respectively). By Step (2) (in our case by [FM99]) both of these
actions can be conjugated to actions by similarities on $\mathbb{R}$. That is,
after conjugation $f_{\gamma}$ scales distances on $\mathbb{R}$ by some
constant $n_{\gamma}$ and $g_{\gamma}$ by some constant $m_{\gamma}$. Note
that these two conjugations together define a quasi-conjugation of the
original quasi-action on SOL. If $n_{\gamma}=m_{\gamma}$ for all
$\gamma\in\Gamma$ then the quasi-action is bounded distance from an action by
isometries. The other case cannot occur because if $n_{\gamma_{0}}\neq
m_{\gamma_{0}}$ for even a single $\gamma_{0}$ then the quasi-isometry
constants of the map induced by $\gamma_{0}^{k}$ go to infinity as
$k\to\infty$ thus violating uniformity of the quasi-action. Now we have that
$\Gamma$ up to finite index and finite kernel acts by isometries on SOL. But
SOL is an index $8$ subgroup of its isometry group so $\Gamma$ must be
virtually a lattice in SOL.

For the proof of Theorem 1.5 we also follow this outline however there are
several extra considerations that apply in our more general setting. First,
our conjugation may only give us an isometric action with respect to a
different metric than the one we started with. Second, the isometry group of a
more general SOL-like group $S$ may be much larger than $S$ (they do not have
the same dimension in general).

Proof of Theorem 1.5. We now prove Theorem 1.5 in detail.

Let $S=(N_{1}\times N_{2})\rtimes\mathbb{R}$ be a SOL-like group equipped with
a left invariant Riemannian metric $g$, where $(N_{i},D_{i})$ is Carnot or
Carnot-by-Carnot. Let $\Gamma$ be a finitely generated group quasi-isometric
to $S$. The quasi-isometry between $\Gamma$ and $S$ induces a quasi-action of
$\Gamma$ on $S$. In particular, each $\gamma\in\Gamma$ gives rise to a
$(L,C)$-quasi-isometry $\phi(\gamma):S\rightarrow S$, where $L\geq 1$, $C\geq
0$ are constants independent of $\gamma$. As explained above, the arguments of
Eskin-Fisher-Whyte imply that $\phi(\gamma)$ is at a bounded distance from a
product map. After replacing $C$ by a larger constant and replacing
$\phi(\gamma)$ by a product map we may assume that $\phi(\gamma)$ is a product
map. After taking an index two subgroup if necessary we may further assume
that each $\phi(\gamma)$ preserves the foliation of $S$ by cosets of
$S_{i}=N_{i}\rtimes_{D_{i}}\mathbb{R}$. Thus for each $\gamma\in\Gamma$, there
are maps $\gamma_{1}:N_{1}\rightarrow N_{1}$, $\gamma_{2}:N_{2}\rightarrow
N_{2}$, $h_{\gamma}:\mathbb{R}\rightarrow\mathbb{R}$, such that $\phi(\gamma)$
is given by:

$\phi(\gamma)(n_{1},n_{2},t)=(\gamma_{1}(n_{1}),\gamma_{2}(n_{2}),h_{\gamma}(t)).$

Let $\tilde{\gamma}_{i}:S_{i}\rightarrow S_{i}$ be given by
$\tilde{\gamma}_{i}(n_{i},t)=(\gamma_{i}(n_{i}),h_{\gamma}(t))$. Then
$\gamma\mapsto\tilde{\gamma}_{i}$ defines a quasi-action of $\Gamma$ on
$S_{i}$. Since this quasi-action is by height-respecting quasi-isometries,
$\Gamma$ induces a uniform quasisimilarity action on $(N_{i},d_{i})$, where
$d_{i}$ is a $D_{i}$-homogeneous distance on $N_{i}$. Due to the particular
form of $\tilde{\gamma}_{i}$ this induced action of $\Gamma$ on $N_{i}$ is
given by $\gamma\mapsto\gamma_{i}$.

Now we assume each $(N_{i},D_{i})$ $(i=1,2$) is either Carnot or Carnot-by-
Carnot with $\dim(\mathfrak{w}_{i})\geq 2$, where $\mathfrak{w}_{i}$ is the
Lie sub-algebra of $\mathfrak{n}_{i}$ generated by the eigenspace of the
smallest eigenvalue of $D_{i}$. Then by Theorem 1.2 (Carnot-by-Carnot case) or
Theorem 3.6 (Carnot case), there is a biLipschitz map $f_{i}$ of $N_{i}$ and a
maximally symmetric $D_{i}$-homogeneous distance $d_{i}$ on $N_{i}$ such that
$f_{i}\circ\gamma_{i}\circ f_{i}^{-1}$ is a similarity of $(N_{i},d_{i})$ for
each $\gamma\in\Gamma$. [KLD17] Theorem 1.2 implies that
$f_{i}\circ\gamma_{i}\circ f_{i}^{-1}$ has the form $f_{i}\circ\gamma_{i}\circ
f_{i}^{-1}=L_{a_{i}}\circ e^{v_{i,\gamma}D_{i}}\circ\phi_{i}$ for some
$a_{i}\in N_{i}$, where $\phi_{i}$ is an automorphism of $N_{i}$ and is also
an isometry of $(N_{i},d_{i})$. .

######  Lemma 4.1.

The equality $v_{1,\gamma}+v_{2,\gamma}=0$ holds for all $\gamma\in\Gamma$.

###### Proof.

Note that the left translation $L_{(0,-v_{1,\gamma})}:S_{1}\rightarrow S_{1}$
(which is an isometry of $S_{1}$) is given by
$L_{(0,-v_{1,\gamma})}(x,t)=(e^{-v_{1,\gamma}D_{1}}x,t-v_{1,\gamma})$. It
follows that $L_{(0,-v_{1,\gamma})}\circ\tilde{\gamma}_{1}$ is a
$(L,C)$-quasi-isometry and is given by
$L_{(0,-v_{1,\gamma})}\circ\tilde{\gamma}_{1}(x,t)=(e^{-v_{1,\gamma}D_{1}}\gamma_{1}(x),h_{\gamma}(t)-v_{1,\gamma})$.
Since the boundary map $e^{-v_{1,\gamma}D_{1}}\circ\gamma_{1}$ of
$L_{(0,-v_{1,\gamma})}\circ\tilde{\gamma}_{1}$ is an isometry, Lemma 5.1 in
[SX12] implies $|h_{\gamma}(t)-v_{1,\gamma}-t|\leq C_{1}$ for a constant
$C_{1}$ that depends only on $L,C$ and $N_{1}$. Similarly by considering
$L_{(0,v_{2,\gamma})}\circ\tilde{\gamma}_{2}$ we get
$|h_{\gamma}(t)+v_{2,\gamma}-t|\leq C_{2}$ for a constant $C_{2}$ that depends
only on $L,C$ and $N_{2}$. Combining these two inequalities we get
$|v_{1,\gamma}+v_{2,\gamma}|\leq C_{1}+C_{2}$ for all $\gamma\in\Gamma$. Since
$v_{i,\gamma^{n}}=nv_{i,\gamma}$ for any $n\geq 1$, the above inequality
applied to $\gamma^{n}$ implies
$|v_{1,\gamma}+v_{2,\gamma}|\leq(C_{1}+C_{2})/n$. Since this is true for all
$n\geq 1$, the lemma follows.

∎

######  Lemma 4.2.

Let $f_{i}:(N_{i},d_{i})\rightarrow(N_{i},d_{i})$ ($i=1,2$) be a biLipschitz
map. Let $d$ be a left invariant Riemannian metric on $S$ such that $N_{1}$,
$N_{2}$ and $\mathbb{R}$ are perpendicular to each other. Define
$F:(S,d)\rightarrow(S,d)$ by $F(n_{1},n_{2},t)=(f_{1}(n_{1}),f_{2}(n_{2}),t)$.
Then $F$ is a $(1,\tilde{C})$-quasi-isometry for some constant $\tilde{C}\geq
0$.

###### Proof.

After rescaling the metric we may assume that vertical lines
$c_{(n_{1},n_{2})}:\mathbb{R}\rightarrow S$,
$c_{(n_{1},n_{2})}(t)=(n_{1},n_{2},t)$, are unit speed geodesics. Let
$\pi_{1}:S\rightarrow S_{1}$ be given by $\pi_{1}(n_{1},n_{2},t)=(n_{1},t)$
and $\pi_{2}:S\rightarrow S_{2}$ be given by
$\pi_{2}(n_{1},n_{2},t)=(n_{2},t)$. Also let $h:S\rightarrow\mathbb{R}$ be the
height function $h(n_{1},n_{2},t)=t$. By Corollary 4.13 of [Fe20] or Theorem
4.1 in [LDPX22] there is a constant $C_{1}\geq 0$ such that for any $p,q\in S$
we have

$|d(p,q)-d^{(1)}(\pi_{1}(p),\pi_{1}(q))-d^{(2)}(\pi_{2}(p),\pi_{2}(q))+|h(p)-h(q)||\leq
C_{1},$

where $d^{(1)}$ is the metric on
$S_{1}\sim(N_{1}\times\\{0\\})\rtimes\mathbb{R}\subset S$ induced by $d$ and
similarly for $d^{(2)}$. Replacing $p,q$ with $F(p)$ and $F(q)$ respectively
we get

$|d(F(p),F(q))-d^{(1)}(\pi_{1}(F(p)),\pi_{1}(F(q)))-d^{(2)}(\pi_{2}(F(p)),\pi_{2}(F(q)))+|h(F(p))-h(F(q))||\leq
C_{1}.$

The lemma follows from the following claim since by the definition of $F$ we
have $h(F(x))=h(x)$ for any $x\in S$.

Claim: There is a constant $\tilde{C}_{i}$ depending only on the Gromov
hyperbolicity constant of $S_{i}$ and the biLipschitz constant of $f_{i}$ such
that for all $p_{1},p_{2}\in S$:
$|d^{(i)}(\pi_{i}(F(p_{1})),\pi_{i}(F(p_{2})))-d^{(i)}(\pi_{i}(p_{1}),\pi_{i}(p_{2}))|\leq\tilde{C}_{i}$.

Proof of the claim: we will only consider the case $i=1$ as the case $i=2$ is
similar. Let $p_{1}=(x_{1},y_{1},t_{1})$, $p_{2}=(x_{2},y_{2},t_{2})$. Then
the claim takes the form

$|d^{(1)}(f_{1}(x_{1}),t_{1}),(f_{1}(x_{2}),t_{2}))-d^{(1)}((x_{1},t_{1}),(x_{2},t_{2}))|\leq\tilde{C}_{1}.$

Let $t_{x_{1},x_{2}}$ be the height at which the two vertical geodesics
$c_{x_{1}}$ and $c_{x_{2}}$ in $S_{1}$ diverge from each other (that is, the
distance between $c_{x_{1}}(t_{x_{1},x_{2}})$ and $c_{x_{2}}(t_{x_{1},x_{2}})$
is $1$), where for $x\in N_{1}$, $c_{x}:\mathbb{R}\rightarrow S_{1}$ is given
by $c_{x}(t)=(x,t)$. Then we have

$|d^{(1)}((x_{1},t_{1}),(x_{2},t_{2}))-(t_{x_{1},x_{2}}-t_{1})-(t_{x_{1},x_{2}}-t_{2})|\leq
C_{2}$

if $t_{x_{1},x_{2}}>\max\\{t_{1},t_{2}\\}$ and
$|d^{(1)}((x_{1},t_{1}),(x_{2},t_{2}))-|t_{1}-t_{2}||\leq C_{2}$ otherwise,
for some constant $C_{2}$ depending only on the Gromov hyperbolicity constant
of $S_{1}$. Similarly

$|d^{(1)}((f_{1}(x_{1}),t_{1}),(f_{1}(x_{2}),t_{2}))-(t_{f_{1}(x_{1}),f_{1}(x_{2})}-t_{1})-(t_{f_{1}(x_{1}),f_{1}(x_{2})}-t_{2})|\leq
C_{2}$

if $t_{f_{1}(x_{1}),f_{1}(x_{2})}>\max\\{t_{1},t_{2}\\}$ and
$|d^{(1)}((f_{1}(x_{1}),t_{1}),(f_{1}(x_{2}),t_{2}))-|t_{1}-t_{2}||\leq C_{2}$
otherwise. It now suffices to show that there is a constant $C_{3}$ such that
$|t_{f_{1}(x_{1}),f_{1}(x_{2})}-t_{x_{1},x_{2}}|\leq C_{3}$ for all
$x_{1},x_{2}\in N_{1}$. This follows from the fact that $d_{1}(x_{1},x_{2})$
is comparable with $e^{t_{x_{1},x_{2}}}$ and that $f_{1}$ is biLipschitz.

∎

Completing the proof of Theorem 1.5. Recall that the $D_{i}$-homogeneous
distance $d_{i}$ on $N_{i}$ is associated to an inner product
$\langle,\rangle_{i}$ on $\mathfrak{n}_{i}$. Since $\phi_{i}$ is an
automorphism of $N_{i}$ and also an isometry of $(N_{i},d_{i})$, Lemma 3.5
implies that $d\phi_{i}$ is layer-preserving and is an orthogonal
transformation with respect to $\langle,\rangle_{i}$. Let
$\langle,\rangle_{0}$ be the inner product on
$T_{e}S=(\mathfrak{n}_{1}\times\mathfrak{n}_{2})\rtimes\mathbb{R}$ that agrees
with $\langle,\rangle_{i}$ on $\mathfrak{n}_{i}$, that satisfies
$\langle(0,0,1),(0,0,1)\rangle_{0}=1$, and such that $\mathfrak{n}_{1}$,
$\mathfrak{n}_{2}$ and $\mathbb{R}$ are perpendicular to each other. Let
$g_{0}$ be the left invariant Riemannian metric on $S$ determined by
$\langle,\rangle_{0}$.

For each $\gamma\in\Gamma$, define a map $\Psi(\gamma):S\rightarrow S$ by

$\Psi(\gamma)(n_{1},n_{2},t)=(f_{1}\circ\gamma_{1}\circ
f_{1}^{-1}(n_{1}),f_{2}\circ\gamma_{2}\circ
f_{2}^{-1}(n_{2}),t+v_{1,\gamma}).$

Since $f_{i}\circ\gamma_{i}\circ f_{i}^{-1}=L_{a_{i}}\circ
e^{v_{i,\gamma}D_{i}}\circ\phi_{i}$, Lemma 4.1 implies
$\Psi(\gamma)(n_{1},n_{2},t)=L_{(a_{1},a_{2},v_{1,\gamma})}(\phi_{1}n_{1},\phi_{2}n_{2},t)$.
The properties of $\phi_{i}$ imply that the map $S\rightarrow S$,
$(n_{1},n_{2},t)\mapsto(\phi_{1}n_{1},\phi_{2}n_{2},t)$ is an automorphism of
$S$ and is an isometry of $(S,g_{0})$. It follows that $\Psi(\gamma)$ is an
isometry of $(S,g_{0})$.

Define $F:S\rightarrow S$ by $F(n_{1},n_{2},t)=(f_{1}(n_{1}),f_{2}(n_{2}),t)$
as in Lemma 4.2. Then $F$ is a quasi-isometry of $S$. Notice that
$\Psi(\gamma)$ and $F\circ\phi(\gamma)\circ F^{-1}$ induce the same boundary
map of $S_{i}$ . It follows that $\Psi(\gamma)$ and $F\circ\phi(\gamma)\circ
F^{-1}$ are at a bounded distance from each other. By replacing
$F\circ\phi(\gamma)\circ F^{-1}$ with $\Psi(\gamma)$ we see that the original
quasi-action of $\Gamma$ on $S$ is now quasi-conjugated to an isometric action
of $\Gamma$ on $(S,g_{0})$. Since the original quasi-action is cobounded, the
isometric action is cocompact and so $\Psi(\Gamma)$ is a uniform lattice in
$\text{Isom}(S,g_{0})$. This finishes the proof when each $(N_{i},D_{i})$
$(i=1,2$) is either Carnot or Carnot-by-Carnot with
$\dim(\mathfrak{w}_{i})\geq 2$.

If $\dim(\mathfrak{w}_{i})=1$ for at least one $i$ and $\Gamma$ is amenable,
then Theorem 1.2 still applies and the above proof works. Now assume
$\dim(\mathfrak{w}_{i})=1$ and $\text{Isom}(S,g)$ admits a uniform lattice
$\Gamma_{0}$ for some left invariant Riemannian metric $g$ on $S$. By Theorem
3 in [W00] $\Gamma_{0}$ is virtually a lattice in some simply connected
solvable Lie group. In particular, $\Gamma_{0}$ is a finitely generated
amenable group. It follows that $\Gamma$ is also amenable, being quasi-
isometric to $\Gamma_{0}$. Hence again Theorem 1.2 applies and the above proof
works.

∎

###  4.2. Quasi-isometric classification of a class of solvable Lie groups

Here we provide the proofs of Theorem 1.8 and Theorem 1.9.

Proof of Theorem 1.8. One direction is clear. For the other direction, we
assume $S_{1},S_{2}\in\mathcal{S}$ are quasi-isometric. Then $S_{1}$ quasi-
acts on $S_{2}$. Since $S_{1}$ is amenable, the proof of Theorem 1.5 applies
(for $\Gamma:=S_{1}$, $S:=S_{2}$) and we conclude that there is a left
invariant Riemannian metric $g_{2}$ on $S_{2}$ such that the quasi-action of
$S_{1}$ on $S_{2}$ is quasi-conjugate to an isometric action of $S_{1}$ on
$(S_{2},g_{2})$. This induces a continuous homomorphism
$\phi:S_{1}\rightarrow\text{Isom}(S_{2},g_{2})$. Since a continuous
homomorphism between Lie groups is a Lie group homomorphism, we see that the
isometric action of $S_{1}$ on $(S_{2},g_{2})$ is a smooth action; that is,
the corresponding map $F:S_{1}\times S_{2}\rightarrow S_{2}$,
$F(s_{1},s_{2})=\phi(s_{1})(s_{2})$, is a smooth map. Fix some $x\in S_{2}$.
Then the map $f:S_{1}\rightarrow S_{2}$ given by
$f(s_{1})=F(s_{1},x)=\phi(s_{1})(x)$ is smooth. It is easy to see that
$\phi(s_{1}^{\prime})\circ f=f\circ L_{s_{1}^{\prime}}$ for all
$s_{1}^{\prime}\in S_{1}$, where $L_{s_{1}^{\prime}}$ is the left translation
of $S_{1}$ by $s_{1}^{\prime}$. This implies the differentials of the map $f$
have constant rank. If this rank is less than $\text{dim}(S_{1})$, then there
exists $h\in f^{-1}(x)\backslash\\{e\\}$. Then all the powers $h^{n}$,
$n\in\mathbb{Z}$, would fix $x\in S_{2}$. This is a contradiction since
$\\{h^{n}|n\in\mathbb{Z}\\}$ is unbounded in $S_{1}$,
$\\{\phi(h^{n})(x)|n\in\mathbb{Z}\\}=\\{x\\}$ is bounded in $S_{2}$, and the
action of $S_{1}$ on $S_{2}$ is quasi-conjugate to the action of $S_{1}$ on
itself by left translations. Hence the rank of $f$ is at least
$\text{dim}(S_{1})$, which implies $\text{dim}(S_{2})\geq\text{dim}(S_{1})$.
By switching the roles of $S_{1}$ and $S_{2}$ we get the reverse inequality
and so $\text{dim}(S_{2})=\text{dim}(S_{1})$. Hence $f$ is a local
diffeomorphism from $S_{1}$ to $S_{2}$. The above argument also shows $f$ is
injective and so is a diffeomorphism onto an open subset of $S_{2}$. Since the
image of $f$ is an orbit of the $S_{1}$ action on $S_{2}$, we see that $f$
must be surjective since otherwise $S_{2}$ would be a disjoint union of at
least two open subsets. Consequently $f$ is a diffeomorphism from $S_{1}$ to
$S_{2}$.

We next use an idea we found in [FLD21], Proposition 2.1. Let $g_{1}$ be the
pullback Riemannian metric of $g_{2}$ by $f$. Then
$f:(S_{1},g_{1})\rightarrow(S_{2},g_{2})$ is an isometry. Finally the equality
$\phi(s_{1}^{\prime})\circ f=f\circ L_{s_{1}^{\prime}}$ implies
$L_{s_{1}^{\prime}}:(S_{1},g_{1})\rightarrow(S_{1},g_{1})$ is an isometry for
any $s_{1}^{\prime}\in S_{1}$. In other words, $g_{1}$ is left invariant.
Hence $S_{1}$ and $S_{2}$ can be made isometric. Since they are both of real
type, Theorem 4.21 in [CKLGO21] implies they are isomorphic.

∎

By using Theorem 1.3 (proved next) and Corollary 1.2, [DFX] instead of Theorem
1.5 the above argument yields Theorem 1.9.

###  4.3. Quasi-isometric rigidity of quasi-actions on certain Heintze groups

Finally we supply the proof of Theorem 1.3.

Proof of Theorem 1.3. Let $(N,D)$ be Carnot-by-Carnot, and $\Gamma$ a group
that quasi-acts coboundedly on $N\rtimes_{D}\mathbb{R}$. We further assume
$\Gamma$ is amenable when $\dim(W)=1$. Then $\Gamma$ induces a uniform
quasisimilarity action on $N$ such that every point in $N$ is a radial limit
point (since the induced action of $\Gamma$ on the space of distinct pairs of
$N$ is cocompact). By Theorem 1.2, there is a biLipschitz map $F_{0}$ of $N$
such that $F_{0}\Gamma F_{0}^{-1}\subset\text{Sim}(N,d_{0})$, where $d_{0}$ is
a fixed maximally symmetric $D$-homogeneous distance on $N$. Recall that the
$D$-homogeneous distance $d_{0}$ is associated with an inner product
$\langle,\rangle_{0}$ on $\mathfrak{n}$. Let $\langle,\rangle$ be the inner
product on $T_{e}(N\rtimes_{D}\mathbb{R})=\mathfrak{n}\times\mathbb{R}$
satisfying
$\langle,\rangle|_{\mathfrak{n}\times\mathfrak{n}}=\langle,\rangle_{0}$,
$\langle\mathfrak{n},\\{0\\}\times\mathbb{R}\rangle=0$ and
$\langle(0,1),(0,1)\rangle=1$. Let $g_{0}$ be the left invariant Riemannian
metric on $N\rtimes_{D}\mathbb{R}$ determined by the inner product
$\langle,\rangle$. Let $\gamma\in\Gamma$. Recall that $F_{0}\gamma F_{0}^{-1}$
acts on $N$ as an affine map and has the form $F_{0}\gamma
F_{0}^{-1}=L_{a_{\gamma}}\circ e^{t_{\gamma}D}\circ\phi_{\gamma}$ for some
$a_{\gamma}\in N$, $t_{\gamma}\in\mathbb{R}$, where $L_{a_{\gamma}}$ is the
left translation of $N$ by $a_{\gamma}$ and $\phi_{\gamma}$ is a graded
automorphism such that $d\phi_{\gamma}$ is a linear isometry of
$(\mathfrak{n},\langle,\rangle_{0})$. Now let $F_{0}\gamma F_{0}^{-1}$ act on
$N\rtimes_{D}\mathbb{R}$ by $F_{0}\gamma
F_{0}^{-1}(n,t)=L_{(a_{\gamma},t_{\gamma})}(\phi_{\gamma}(n),t)$, where
$L_{(a_{\gamma},t_{\gamma})}$ is the left translation of
$N\rtimes_{D}\mathbb{R}$ by $(a_{\gamma},t_{\gamma})$. It is easy to check
that this defines an isometric action of $\Gamma$ on
$(N\rtimes_{D}\mathbb{R},g_{0})$ that induces the given action of $F_{0}\Gamma
F_{0}^{-1}$ on $N=\partial S\backslash\\{\infty\\}$.

∎

##  5\. BiLipschitz maps of diagonal Heintze pairs

In this section we study individual biLipschitz maps of diagonal Heintze pairs
(see section 3.4). We show that if a biLipschitz map permutes the cosets of a
connected graded normal subgroup, then it has an expression (we call it
compatible expression) with some nice properties, see Lemma 5.4.

For any biLipschitz map $F:N\rightarrow N$ and $g\in N$, we denote by
$F_{g}:=L_{F(g)^{-1}}\circ F\circ L_{g}$. Note that $F_{g}(0)=0$.

######  Lemma 5.1.

Let $(\mathfrak{n},D)$ be a diagonal Heintze pair and $\mathfrak{w}$ an ideal
of $\mathfrak{n}$ such that $D(\mathfrak{w})=\mathfrak{w}$. Let
$F:N\rightarrow N$ be a biLipschitz map that permutes the cosets of $W$, where
$W$ is the connected Lie subgroup of $N$ with Lie algebra $\mathfrak{w}$. In
addition assume for every $g\in N$, the map $F_{g}|_{W}:W\to W$ is an
automorphism of $W$. Then there is an automorphism $\phi:W\rightarrow W$ such
that $d\phi:\mathfrak{w}\rightarrow\mathfrak{w}$ is layer-preserving,
$F(gw)=F(g)\phi(w)$ and
$(\chi_{{F_{0}(g)}}|_{W})\circ\phi=\phi\circ(\chi_{g}|_{W})$ hold for any
$g\in N$, where $\chi_{x}$ denotes the conjugation by $x$.

######  Remark 5.2.

If we start with a uniform quasisimilarity group $\Gamma$ of a Carnot-by-
Carnot group $N$ and $W$ is the connected Lie subgroup of $N$ with Lie algebra
generated by the eigenspace of the smallest eigenvalue of $D$, we can apply
the fiber Tukia theorem (Theorem 3.7) when $\dim(W)\geq 2$ such that every
element in a biLipschitz conjugate of $\Gamma$ satisfies the assumption in the
lemma.

###### Proof.

By replacing $F$ with $F_{0}$ we may assume $F(0)=0$. Since $W$ is normal in
$N$, $gW=Wg$ for any $g\in N$. The fact that $F$ permutes the cosets of $W$
implies that $F(gW)=F(g)W$ and $F(Wg)=WF(g)$. Hence there are two functions
$\phi_{g},\psi_{g}:W\rightarrow W$ such that $F(gw)=F(g)\phi_{g}(w)$ and
$F(wg)=\psi_{g}(w)F(g)$ for any $w\in W$. Notice that $\phi_{g}=F_{g}|_{W}$
and so by assumption is an automorphism of $W$. By definition
$\psi_{g}(w)=F(wg)F(g)^{-1}=F(g\cdot
g^{-1}wg)F(g)^{-1}=F(g)\phi_{g}(g^{-1}wg)F(g)^{-1}$. It follows that

(4) $\psi_{g}=(\chi_{F(g)}|_{W})\circ\phi_{g}\circ(\chi_{g^{-1}}|_{W})$

is a composition of automorphisms of $W$ and so is an automorphism of $W$.
Here we used the assumption that $W$ is normal in $N$ and so $\chi_{x}|_{W}$
is an automorphism of $W$ for any $x\in N$.

We claim that $\psi:=\psi_{g}$ is independent of $g$. Let $g_{1},g_{2}\in N$.
Then for any $w\in W$, $d(wg_{1},wg_{2})=d(g_{1},g_{2})$. Since $F$ is
$L$-biLipschitz for some $L\geq 1$, we have $d(F(wg_{1}),F(wg_{2}))\leq
Ld(g_{1},g_{2})$. Since $F(wg_{i})=\psi_{g_{i}}(w)F(g_{i})$, we get

$\displaystyle d(\psi_{g_{1}}(w),\psi_{g_{2}}(w))$ $\displaystyle\leq
d(\psi_{g_{1}}(w),F(wg_{1}))+d(F(wg_{1}),F(wg_{2}))+d(F(wg_{2}),\psi_{g_{2}}(w))$
$\displaystyle\leq d(0,F(g_{1}))+Ld(g_{1},g_{2})+d(F(g_{2}),0)$

for all $w\in W$. By Lemma 3.1 we have $\psi_{g_{1}}=\psi_{g_{2}}$.

We now show that $\phi:=\phi_{g}$ is also independent of $g$. Since
$D(\mathfrak{w})=\mathfrak{w}$, the restriction of a $D$-homogeneous distance
to $W$ is a $D|_{\mathfrak{w}}$-homogeneous distance on $W$. As the
restriction of a biLipschitz map, the automorphism $\phi_{g}$ is a biLipschitz
map of $W$. By Lemma 3.5, $d\phi_{g}:\mathfrak{w}\rightarrow\mathfrak{w}$ is
layer preserving. Write $\mathfrak{w}=W_{\mu_{1}}\oplus\cdots\oplus
W_{\mu_{m}}$ as the direct sum of eigenspaces of $D|_{\mathfrak{w}}$, where
$0<\mu_{1}<\cdots<\mu_{m}$ are the distant eigenvalues of $D|_{\mathfrak{w}}$.
Then $d\phi_{g}(W_{\mu_{j}})=W_{\mu_{j}}$. Let $g,h\in N$. We shall show that
for each $j$ the equality $d\phi_{g}|_{W_{\mu_{j}}}=d\phi_{h}|_{W_{\mu_{j}}}$
holds. To see this, notice that for any $i$ and $x\in N$, if we denote
$I_{i}=\oplus_{k\geq i}W_{\mu_{k}}$, then (2) implies $d\chi_{x}(I_{i})\subset
I_{i}$ and that $d\chi_{x}$ induces the identity map on $I_{j}/{I_{j+1}}\cong
W_{\mu_{j}}$. Hence the map on $I_{j}/{I_{j+1}}$ induced by
$d(\chi_{F(g)}|_{W})\circ d\phi_{g}\circ d(\chi_{g^{-1}}|_{W})$ equals
$d\phi_{g}|_{W_{\mu_{j}}}$. The same is true when $g$ is replaced by $h$. Now
(4) and the fact that $d\psi_{g}=d\psi_{h}$ implies
$d\phi_{g}|_{W_{\mu_{j}}}=d\phi_{h}|_{W_{\mu_{j}}}$.

Finally, by picking $g=0$ in (4) we obtain $\phi=\psi$ as $F(0)=0$. The
equality $(\chi_{{F_{0}(g)}}|_{W})\circ\phi=\phi\circ(\chi_{g}|_{W})$ for any
$g\in N$ also follows from (4).

∎

Below when we say $F:\mathfrak{n}\rightarrow\mathfrak{n}$ is a biLipschitz
map, we identify $\mathfrak{n}$ with $N$ and $\mathfrak{n}$ is equipped with a
$D$-homogeneous distance.

A self homeomorphism $F:G\to G$ of a Lie group is called an affine map if
$F=L_{g}\circ\phi$, where $\phi$ is an automorphism of $G$ and $L_{g}$ is left
translation by $g\in G$. We say $\phi$ is the automorphism part of the affine
map $F$.

Let $(\mathfrak{n},D)$ and $\mathfrak{w}$ be as in Lemma 5.1. Denote by
$\bar{D}:{\mathfrak{n}}/{\mathfrak{w}}\rightarrow{\mathfrak{n}}/{\mathfrak{w}}$
the derivation of ${\mathfrak{n}}/{\mathfrak{w}}$ induced by $D$ and
$\sigma(\bar{D})$ the set of eigenvalues of $\bar{D}$. Let
$H\subset\mathfrak{n}$ be a graded subspace of $\mathfrak{n}$ complementary to
$\mathfrak{w}$; that is, for each $\lambda\in\sigma(\bar{D})$,
$H_{\lambda}\subset V_{\lambda}$ is a complementary linear subspace of
$W_{\lambda}$ in $V_{\lambda}$ (if $W_{\lambda}=\\{0\\}$, then
$H_{\lambda}=V_{\lambda}$), and
$H=\oplus_{\lambda\in\sigma(\bar{D})}H_{\lambda}$. Notice that for every
$g\in\mathfrak{n}$, there are unique $h\in H$, $w\in\mathfrak{w}$ such that
$g=h*w$.

For $x\in\mathfrak{n}$, we use $\bar{x}$ to denote $\pi(x)$, where
$\pi:\mathfrak{n}\to\mathfrak{n}/\mathfrak{w}$ is the quotient map. For
convenience, we introduce the following terminology.

######  Definition 5.3.

Let $(\mathfrak{n},D)$ be a diagonal Heintze pair and $\mathfrak{w}$ an ideal
of $\mathfrak{n}$ such that $D(\mathfrak{w})=\mathfrak{w}$. Let
$F:N\rightarrow N$ be a biLipschitz map that permutes the cosets of $W$, where
$W$ is the connected Lie subgroup of $N$ with Lie algebra $\mathfrak{w}$. In
addition assume that $F$ induces an affine map of $N/W$ and that for every
$g\in N$, the map $F_{g}|_{W}:W\to W$ is an automorphism of $W$. Let $\bar{B}$
be the automorphism part of the affine map of $N/W$ induced by $F$. We say an
expression $F(h*w)=F(0)*Bh*Aw*As(\bar{h})$ ($h\in H$, $w\in\mathfrak{w}$),
with $A=d\phi$, where $\phi$ is the automorphism of $W$ from Lemma 5.1, is a
compatible expression of $F$ if the following hold:
1\. $B:H\rightarrow\mathfrak{n}$ is a linear map satisfying
$B(H_{\lambda})\subset V_{\lambda}$ and $d\bar{B}\circ\pi|_{H}=\pi\circ B$,
where $\pi:\mathfrak{n}\rightarrow\bar{\mathfrak{n}}$ is the quotient map;
2\. $[Bh,Aw]=A[h,w]$ for any $h\in H$, $w\in\mathfrak{w}$;
3\. $s$ is a map $\mathfrak{n}/\mathfrak{w}\rightarrow Z(\mathfrak{w})$, where
$Z(\mathfrak{w})$ denotes the center of $\mathfrak{w}$.

We shall show (see Lemma 5.5) that conditions 1-2 above imply condition 3. But
to prove this we need the existence of a compatible expression (Lemma 5.4).

In general a biLipschitz map $F$ does not have a unique compatible expression.
The reason is that there might be more than one linear map
$B:H\rightarrow\mathfrak{n}$ satisfying Conditions 1 and 2 above. As a result,
the map $s$ is also not unique. However, if $\alpha$ denotes the smallest
eigenvalue of $\bar{D}$ and $\pi_{\lambda}:\mathfrak{w}\rightarrow
W_{\lambda}$ is the projection with respect to the decomposition
$\mathfrak{w}=\oplus_{\lambda}W_{\lambda}$, then $\pi_{\lambda}\circ s$ for
$\lambda<\alpha$ is unique since a change in $B$ only affects
$\pi_{\lambda}\circ s$ for $\lambda\geq\alpha$. Another way to see this is to
notice $(\pi_{\lambda}\circ s)(\bar{h})=A^{-1}(\pi_{\lambda}(F(0)^{-1}*F(h)))$
for $\lambda<\alpha$.

######  Lemma 5.4.

Let $(\mathfrak{n},D)$, $\mathfrak{w}$ and $F$ be as in Definition 5.3. Then
$F$ has a compatible expression.

The proof of Lemma 5.4 is tedious. To improve readability we have placed the
proof of Lemma 5.4 in Appendix C. The main ingredient in the proof is the fact
that $\phi\circ\chi_{g}=\chi_{G(g)}\circ\phi$, where $G=F_{0}$. See Lemma 5.1.
We also implicitly (and repeatedly) use the fact that
$[Z(\mathfrak{w}),\mathfrak{n}]\subset Z(\mathfrak{w})$. This follows from the
Jacobi identity and the fact that $\mathfrak{w}$ is an ideal of
$\mathfrak{n}$.

Notice that the order of appearance of $Aw$ and $As(\bar{h})$ in (1) and (2)
below are different.

######  Lemma 5.5.

Let $(\mathfrak{n},D)$, $\mathfrak{w}$ and $F$ be as in Definition 5.3. Using
the notation $p=h*w\in\mathfrak{n}$ where $h\in H$, $w\in\mathfrak{w}$:
(1) If an expression $F(h*w)=F(0)*Bh*Aw*As(\bar{h})$ for $F$ satisfies
conditions 1-2 of Definition 5.3, then it also satisfies condition 3;
(2) If an expression $F(h*w)=F(0)*Bh*As(\bar{h})*Aw$ for $F$ satisfies
conditions 1-2 of Definition 5.3, then it also satisfies condition 3
($s(\bar{h})\in Z(\mathfrak{w})$) and so $F(h*w)=F(0)*Bh*Aw*As(\bar{h})$ is a
compatible expression.

###### Proof.

(1) Let $F(h*w)=F(0)*B_{1}h*Aw*As_{1}(\bar{h})$ be a compatible expression.
Condition 1 implies $B_{1}h-Bh\in\mathfrak{w}$ for all $h\in H$. From
condition 2 we have $[Bh,Aw]=A[h,w]=[B_{1}h,Aw]$ for all $h\in H$ and all
$w\in\mathfrak{w}$. It follows that $[B_{1}h-Bh,Aw]=0$ and so
$s^{\prime}(h):=B_{1}h-Bh\in Z(\mathfrak{w})$. By using the BCH formula and
the fact that $[\mathfrak{n},Z(\mathfrak{w})]\subset Z(\mathfrak{w})$ we get
$(-Bh)*B_{1}h\in Z(\mathfrak{w})$. Finally from the two expressions for $F(h)$
we get $As(\bar{h})=(-Bh)*B_{1}h*As_{1}(\bar{h})\in Z(\mathfrak{w})$.

(2) The same proof as above also shows in this case $s(\bar{h})\in
Z(\mathfrak{w})$. This allows us to switch $Aw$ and $As(\bar{h})$ to get a
compatible expression.

∎

We observe that if $[Bh,Aw]=A[h,w]$ for all $h\in H$, $w\in\mathfrak{w}$, then

(5)
$Bh*Aw*(Bh)^{-1}=\sum_{i=0}^{\infty}\frac{1}{i!}(\text{ad}\,(Bh))^{i}(Aw)=A(\sum_{i=0}^{\infty}\frac{1}{i!}(\text{ad}\,(h))^{i}(w))=A(h*w*h^{-1}).$

For later use (in Lemma 8.8), we record the following lemma. The main point of
the lemma is that the same linear map $B$ works for $F_{p}$.

######  Lemma 5.6.

Let $F:\mathfrak{n}\rightarrow\mathfrak{n}$ be a biLipschitz map with a
compatible expression $F(h*w)=F(0)*Bh*Aw*As(\bar{h})$. Then for any $p\in N$,
the map $F_{p}$ admits a compatible expression of the form
$F_{p}(h*w)=Bh*Aw*A\tilde{s}(\bar{h})$ for some map
$\tilde{s}:\mathfrak{n}/\mathfrak{w}\rightarrow Z(\mathfrak{w})$.

###### Proof.

Note that $(F_{p})_{q}=F_{pq}$ for any $p,q\in N$ so the same $A$ works for
$F_{p}$. By Lemma 5.1 we can write $F_{p}(h*w)=F_{p}(h)*Aw$. Since $F_{p}$
induces the automorphism $\bar{B}$ on $N/W$, we can write
$F_{p}(h)=Bh*A\tilde{s}(\bar{h})$ for some map
$\tilde{s}:\bar{\mathfrak{n}}\rightarrow\mathfrak{w}$. So we have
$F_{p}(h*w)=Bh*A\tilde{s}(\bar{h})*Aw$ with Conditions 1 and 2 of Definition
5.3 satisfied. By Lemma 5.5 (2) $F_{p}(h*w)=Bh*Aw*A\tilde{s}(\bar{h})$ is a
compatible expression for $F_{p}$.

∎

##  6\.  Characterization of biLipschitz shear maps

In Sections 6–8 and Appendix C we shall often identify a simply connected
nilpotent Lie group with its Lie algebra via the exponential map. (See Section
3.2.)

From now on, unless stated otherwise, $(N,D)$ will be Carnot-by-Carnot and $W$
will be the Lie subgroup of $N$ with Lie algebra $\mathfrak{w}$ generated by
the eigenspace of the smallest eigenvalue of $D$. In this section we will
characterize continuous maps $s:N/W\rightarrow W$ such that the shear maps
$F:(N,d)\rightarrow(N,d)$ given by $F(g)=g\,s(gW)$ are biLipschitz.
BiLipschitz shear maps will be used to conjugate a uniform quasisimilarity
group into a similarity group. We are only interested in the case $s(0)=0$. In
this case, $F(0)=0$. We first introduce a useful differential one form on
Carnot groups, which will be used in the characterization of biLipschitz shear
maps.

###  6.1. Horizontal tautological one form on Carnot groups

We first recall the tautological one form on Lie groups and then define the
horizontal tautological one form on Carnot groups. Finally we give an
expression for the horizontal tautological one form in exponential
coordinates.

For a Lie group $G$ with Lie algebra $\mathfrak{g}=T_{e}G$, the tautological
one form $\theta$ is a $\mathfrak{g}$ valued left invariant one form on $G$
defined as follows. For each $x\in G$,
$\theta_{x}:T_{x}G\rightarrow\mathfrak{g}$ is the linear map given by
$\theta_{x}(X)=\tilde{X}_{e}$, where $\tilde{X}$ is the left invariant vector
field on $G$ satisfying $\tilde{X}_{x}=X$ and $\tilde{X}_{g}$ denotes the
value of $\tilde{X}$ at $g\in G$. The name “tautological” comes from the fact
that at each $x\in G$, $\theta_{x}$ is the identity map if one identifies the
tangent vectors with the corresponding left invariant vector fields.

The horizontal tautological one form is a counterpart of tautological one form
in the setting of Carnot groups. Let $G$ be a Carnot group with Lie algebra
grading $\mathfrak{g}=\oplus_{j=1}^{n}V_{j}$. Then the horizontal tautological
one form $\theta_{H}$ is a $V_{1}$-valued left invariant one form on $G$. Let
$\pi_{1}:\mathfrak{g}\rightarrow V_{1}$ be the projection with respect to the
decomposition $\mathfrak{g}=\oplus_{j=1}^{n}V_{j}$. Define
$\theta_{H}=\pi_{1}\circ\theta$; that is, for each $x\in G$,
$(\theta_{H})_{x}:T_{x}G\rightarrow V_{1}$ is defined to be the composition
$\pi_{1}\circ\theta_{x}$.

Let $e_{1},\cdots,e_{k_{n}}$ be a basis of $\mathfrak{g}=T_{e}G$ such that
$\\{e_{1},\cdots,e_{k_{1}}\\}$ is a basis of $V_{1}$. For each $i$, let
$X_{i}$ be the left invariant vector field on $G$ that equals $e_{i}$ at $e$.
Let $\\{\theta_{1},\cdots,\theta_{k_{n}}\\}$ be the basis of the space of left
invariant one forms on $G$ dual to $\\{e_{1},\cdots,e_{k_{n}}\\}$. So we have
$\theta_{i}(X_{j})=\delta_{ij}$. Now it is easy to see that
$\theta_{H}=\sum_{i=1}^{k_{1}}\theta_{i}e_{i}$ as
$(\sum_{i=1}^{k_{1}}\theta_{i}e_{i})(\sum_{j=1}^{k_{n}}a_{j}X_{j})=\sum_{i=1}^{k_{1}}a_{i}e_{i}$.
If we use the exponential coordinates
$\mathbb{R}^{\dim\mathfrak{g}}\rightarrow G$,
$(x_{1},\cdots,x_{k_{n}})\mapsto\text{exp}(\sum x_{j}e_{j})$, then
$\theta_{i}$ is given by $\theta_{i}=dx_{i}$ for $1\leq i\leq k_{1}$. Hence
$\theta_{H}$ has the expression $\theta_{H}=\sum_{i=1}^{k_{1}}dx_{i}e_{i}$.

###  6.2. Structure of Lie algebra when $\alpha$ is not an integer

We return to the case that $(N,D)$ is Carnot-by-Carnot. We rescale so that the
smallest eigenvalue of $D$ is $1$. Let $\alpha>1$ be the smallest eigenvalue
of the induced derivation
$\bar{D}:{\mathfrak{n}}/{\mathfrak{w}}\rightarrow{\mathfrak{n}}/{\mathfrak{w}}$.
Under the assumption that $\alpha$ is not an integer we clarify the algebraic
structure of $\mathfrak{n}$ and show that $\mathfrak{n}$ is a central product.

Recall
$\bar{\mathfrak{n}}=\mathfrak{n}/\mathfrak{w}=\bar{V}_{1}\oplus\cdots\oplus\bar{V}_{m}$
is assumed to be a Carnot algebra and
$\pi:\mathfrak{n}\rightarrow\bar{\mathfrak{n}}$ is the quotient map.

######  Lemma 6.1.

(1) If $\alpha>1$ is irrational, then $\mathfrak{n}$ has an ideal
$\mathfrak{h}$ that is mapped by $\pi$ isomorphically onto
$\bar{\mathfrak{n}}$. In particular, $\mathfrak{n}$ is the direct sum of two
ideals $\mathfrak{w}$ and $\mathfrak{h}$.
(2) Suppose $\alpha>1$ is rational but not an integer. Let
$k_{0}=\min\\{k\in\mathbb{N}|\,k\alpha\;\;\text{is an integer}\\}$. Then there
exist a graded central ideal $I$ of $\mathfrak{w}$ contained in
$\oplus_{l}W_{lk_{0}\alpha}$, a Carnot algebra $\mathfrak{h}=\oplus_{j}H_{j}$
with a graded central ideal $J$ contained in $\oplus_{l}H_{lk_{0}}$, and a
linear isomorphism $\phi:I\rightarrow J$ satisfying $\phi(I\cap
W_{lk_{0}\alpha})=J\cap H_{lk_{0}}$, such that $\mathfrak{n}$ is isomorphic to
the central product
$\mathfrak{w}\times_{\phi}{\mathfrak{h}}=(\mathfrak{w}\oplus{\mathfrak{h}})/K$,
where $K=\\{(x,\phi(x))|x\in I\\}$.

###### Proof.

Let $H_{1}=V_{\alpha}$ and $\mathfrak{h}$ be the Lie subalgebra of
$\mathfrak{n}$ generated by $H_{1}$. Then $\pi(H_{1})=\bar{V}_{1}$ and
$\pi(\mathfrak{h})=\bar{\mathfrak{n}}$. Furthermore, the property
$[V_{a},V_{b}]\subset V_{a+b}$ implies $\mathfrak{h}$ is a Carnot algebra.
(1) Assume $\alpha$ is irrational. Since
$\mathfrak{h}\subset\oplus_{l}V_{l\alpha}$ and
$\mathfrak{w}\subset\oplus_{j\geq 1}V_{j}$, we have
$\mathfrak{w}\cap\mathfrak{h}=\\{0\\}$. As $\mathfrak{w}$ is an ideal of
$\mathfrak{n}$, the property $[V_{a},V_{b}]\subset V_{a+b}$ implies
$[\mathfrak{w},\mathfrak{h}]=0$. It follows that $\mathfrak{h}$ is an ideal of
$\mathfrak{n}$ and $\mathfrak{n}=\mathfrak{w}\oplus\mathfrak{h}$ is a direct
sum of two ideals.

(2) Suppose $\alpha>1$ is rational but not an integer. The fact
$[H_{1},W_{1}]\subset V_{1+\alpha}\cap\mathfrak{w}=\\{0\\}$ implies
$[\mathfrak{w},\mathfrak{h}]=0$ and so $\mathfrak{h}$ is also an ideal of
$\mathfrak{n}$. However in this case $\mathfrak{h}$ and $\mathfrak{w}$ may
have nontrivial intersection. The fact $[\mathfrak{w},\mathfrak{h}]=0$ implies
$\mathfrak{w}\cap\mathfrak{h}$ is central in $\mathfrak{n}$ and so is central
in both $\mathfrak{w}$ and $\mathfrak{h}$. Define
$f:\mathfrak{w}\oplus\mathfrak{h}\rightarrow\mathfrak{n}$ by $f(w,h)=w+h$.
Then $f$ is a surjective Lie algebra homomorphism with kernel
$f^{-1}(0)=\\{(w,h)|w\in\mathfrak{w}\cap\mathfrak{h},h=-w\\}$. Set
$I=J=\mathfrak{w}\cap\mathfrak{h}$ and define $\phi:I\rightarrow J$ by
$\phi(w)=-w$. Then $\mathfrak{n}\cong\mathfrak{w}\times_{\phi}\mathfrak{h}$.
Finally as the intersection of two graded ideals $\mathfrak{w}$,
$\mathfrak{h}$ of $\mathfrak{n}$, $\mathfrak{w}\cap\mathfrak{h}$ ($=I$=$J$) is
a graded ideal in both $\mathfrak{w}$ and $\mathfrak{h}$.

∎

Since $[\mathfrak{w},\mathfrak{h}]=0$ and
$\mathfrak{n}=\mathfrak{w}+\mathfrak{h}$, we see that $Z(\mathfrak{w})$ lies
in the center of $\mathfrak{n}$.

###  6.3. Characterization of biLipschitz shear maps

Let $s:N/W\rightarrow W$ be a map satisfying $s(0)=0$ and
$F:(N,d)\rightarrow(N,d)$ be given by $F(g)=gs(gW)$.

######  Lemma 6.2.

Assume $F$ as above is biLipschitz. Then $s$ takes values in the center $Z(W)$
of $W$.

###### Proof.

As $s(gW)\in W$, $F$ maps each coset of $W$ to itself. Let $g\in N$ and
$A_{g}:W\to W$ be given by $A_{g}=F_{g}|_{W}$. Observe that $A_{g}$ is an
inner automorphism of $W$: $A_{g}(w)=s(gW)^{-1}ws(gW)$. As $F$ is biLipschitz,
so is $A_{g}$. A biLipschitz automorphism of a Carnot group is necessarily
graded (see Lemma 3.5) and so is completely determined by its action on the
first layer. On the other hand, an inner automorphism induces the trivial map
on $\mathfrak{w}/[\mathfrak{w},\mathfrak{w}]$. It follows that $A_{g}$ is the
trivial automorphism and therefore $s(gW)$ must lie in $Z(W)$.

∎

We need to introduce some function spaces before we can give the statement of
the characterization.

The spaces $\mathcal{H}_{j}$. Let
$\pi:\mathfrak{n}\rightarrow\mathfrak{n}/\mathfrak{w}$ be the natural
projection. For any $\bar{X}\in\mathfrak{n}/\mathfrak{w}$ and $z\in
Z(\mathfrak{w})$, we define $[\bar{X},z]:=[X,z]$, where $X\in\mathfrak{n}$ is
such that $\pi(X)=\bar{X}$. This is well-defined since $z\in Z(\mathfrak{w})$
and different choices of $X$ differ by an element in $\mathfrak{w}$. Notice
$[\bar{X},z]\in Z(\mathfrak{w})$.

Let $\theta_{H}$ be the horizontal tautological one form on
$\mathfrak{n}/\mathfrak{w}$. For any $j\geq 1$, denote by $P_{j}$ the space of
all continuous maps $c:\mathfrak{n}/\mathfrak{w}\rightarrow
Z_{j}(\mathfrak{w})$ (recall $Z_{j}(\mathfrak{w})=Z(\mathfrak{w})\cap W_{j}$)
satisfying $c(0)=0$ and $\int_{\gamma}[c(x),\theta_{H}(x)]=0$ for all closed
horizontal curves $\gamma$ in $\mathfrak{n}/\mathfrak{w}$. Here we are using
our definition of bracket $[z,\bar{X}]$ for $z\in Z(\mathfrak{w})$,
$\bar{X}\in\mathfrak{n}/\mathfrak{w}$. Notice $[c(x),\theta_{H}(x)]\in
Z_{j+\alpha}(\mathfrak{w})$ as $c(x)\in Z_{j}(\mathfrak{w})$ and
$\theta_{H}(x)\in\bar{V}_{1}$. For any $c\in P_{j}$, define a map
$c^{(1)}:\mathfrak{n}/\mathfrak{w}\rightarrow Z_{j+\alpha}(\mathfrak{w})$ by
$c^{(1)}(p)=\int_{\gamma}[c(x),\theta_{H}(x)]$, where $\gamma$ is any
horizontal path from $0$ to $p$. This is well-defined by the definition of
$P_{j}$. If $c^{(1)}\in P_{j+\alpha}$, then we define
$c^{(2)}=(c^{(1)})^{(1)}$. Similarly, we can define
$c^{(k)}:\mathfrak{n}/\mathfrak{w}\rightarrow Z_{j+k\alpha}(\mathfrak{w})$ if
$c^{(k-1)}\in P_{j+(k-1)\alpha}$.

For each integer $1\leq j\leq\alpha$, let $E_{j}$ be the space of
$\frac{j}{\alpha}$ -Holder continuous maps
$c:\mathfrak{n}/\mathfrak{w}\rightarrow Z_{j}(\mathfrak{w})$ satisfying
$c(0)=0$, and $\mathcal{H}_{j}$ be the set of elements $c\in E_{j}$ such that
$c^{(k)}$ is defined for all $k\geq 1$ (i.e., $c^{(k-1)}\in
P_{j+(k-1)\alpha}$). Here the metric on $\mathfrak{n}/\mathfrak{w}$ is a
Carnot metric and the metric on $Z_{j}(\mathfrak{w})$ is an Euclidean metric.
Note that $c^{(k)}\equiv 0$ for large enough $k$ since $\mathfrak{n}$ is
nilpotent.

Special case. We notice that $\mathcal{H}_{j}=E_{j}$ for all $1\leq j<\alpha$
when $\alpha$ is not an integer. This is because in this case
$[Z(\mathfrak{w}),\mathfrak{n}]=0$ and so $c^{(i)}\equiv 0$ for all $c\in
E_{j}$ and all $i\geq 1$.

Given a continuous map $s:\mathfrak{n}/\mathfrak{w}\rightarrow
Z(\mathfrak{w})$ satisfying $s(0)=0$, define
$K:\mathfrak{n}/\mathfrak{w}\times\mathfrak{n}/\mathfrak{w}\rightarrow
Z(\mathfrak{w})$ by

$K(\bar{g}_{1},\bar{g}_{2})=s(\bar{g}_{2})*(g_{1}^{-1}*g_{2})^{-1}*(-s(\bar{g}_{1}))*(g_{1}^{-1}*g_{2}).$

Notice that $K$ is well-defined as $s$ takes values in $Z(\mathfrak{w})$.
Since both $s(\bar{g}_{2})$ and
$(g_{1}^{-1}*g_{2})^{-1}*(-s(\bar{g}_{1}))*(g_{1}^{-1}*g_{2})$ lie in
$Z(\mathfrak{w})$, the BCH formula implies
$K(\bar{g}_{1},\bar{g}_{2})=s(\bar{g}_{2})+(g_{1}^{-1}*g_{2})^{-1}*(-s(\bar{g}_{1}))*(g_{1}^{-1}*g_{2})$.

######  Lemma 6.3.

Write $s=\sum_{j}s_{j}$, where $s_{j}:\mathfrak{n}/\mathfrak{w}\rightarrow
Z_{j}(\mathfrak{w})$ is the $W_{j}$ component of $s$.
(1) Assume $\alpha$ is not an integer. If there is a constant $C>0$ such that
$|\pi_{j}(K(\bar{g}_{1},\bar{g}_{2}))|^{\frac{1}{j}}\leq
Cd^{\frac{1}{\alpha}}_{CC}(\bar{g}_{1},\bar{g}_{2})$ for all $j\geq 1$ and all
$\bar{g}_{1},\bar{g}_{2}\in\mathfrak{n}/\mathfrak{w}$, then $s_{j}\in E_{j}$
for $1\leq j<\alpha$ and $s_{j}\equiv 0$ for $j>\alpha$;
(2) Assume $\alpha$ is an integer and $j_{0}$ is an integer satisfying $1\leq
j_{0}\leq\alpha$. If there is a constant $C>0$ such that
$|\pi_{k\alpha+j_{0}}(K(\bar{g}_{1},\bar{g}_{2}))|^{\frac{1}{k\alpha+j_{0}}}\leq
Cd^{\frac{1}{\alpha}}_{CC}(\bar{g}_{1},\bar{g}_{2})$ for all $k\geq 0$ and all
$\bar{g}_{1},\bar{g}_{2}\in\mathfrak{n}/\mathfrak{w}$, then
$s_{j_{0}}\in\mathcal{H}_{j_{0}}$ and $s_{k\alpha+j_{0}}=s_{j_{0}}^{(k)}$ for
all $k\geq 1$.

###### Proof.

Notice

$\displaystyle
K(\bar{g}_{1},\bar{g}_{2})=s(\bar{g}_{2})-s(\bar{g}_{1})+[-(g_{1}^{-1}*g_{2}),-s(\bar{g}_{1})]+\sum_{k\geq
2}\frac{1}{k!}(ad(-(g_{1}^{-1}*g_{2})))^{k}(-s(\bar{g}_{1})).$

Since $s$ takes values in $Z(\mathfrak{w})$, using our definition of bracket
$[\bar{X},z]$ with $\bar{X}\in\mathfrak{n}/\mathfrak{w}$, $z\in
Z(\mathfrak{w})$, we can write

$K(\bar{g}_{1},\bar{g}_{2})=s(\bar{g}_{2})-s(\bar{g}_{1})+[-(\bar{g}_{1}^{-1}*\bar{g}_{2}),-s(\bar{g}_{1})]+\sum_{k\geq
2}\frac{1}{k!}(ad(-(\bar{g}_{1}^{-1}*\bar{g}_{2})))^{k}(-s(\bar{g}_{1})).$

(1) Assume $\alpha$ is not an integer. Since
$[Z(\mathfrak{w}),\mathfrak{n}]=0$ in this case we have
$K(\bar{g}_{1},\bar{g}_{2})=s(\bar{g}_{2})-s(\bar{g}_{1})$. The assumption
implies $|s_{j}(\bar{g}_{2})-s_{j}(\bar{g}_{1})|^{\frac{1}{j}}\leq
Cd^{\frac{1}{\alpha}}_{CC}(\bar{g}_{1},\bar{g}_{2})$ for all $j\geq 1$ and all
$\bar{g}_{1},\bar{g}_{2}\in\mathfrak{n}/\mathfrak{w}$. When $1\leq j<\alpha$,
this implies $s_{j}\in E_{j}$. When $j>\alpha$, this implies $s_{j}$ is a
constant function and so $s_{j}\equiv 0$ as $s_{j}(0)=0$.

(2) Assume $\alpha$ is an integer. As
$[-(\bar{g}_{1}^{-1}*\bar{g}_{2}),-s(\bar{g}_{1})]\in\oplus_{i\geq\alpha+1}W_{i}$,
we have
$\pi_{j}K(\bar{g}_{1},\bar{g}_{2})=s_{j}(\bar{g}_{2})-s_{j}(\bar{g}_{1})$ for
$1\leq j\leq\alpha$. The assumption implies $s_{j_{0}}\in E_{j_{0}}$. Let
$\bar{\gamma}:[a,b]\rightarrow\mathfrak{n}/\mathfrak{w}$ be a rectifiable
horizontal curve parametrized by arc length. Then
$d_{CC}(\bar{\gamma}(t_{1}),\bar{\gamma}(t_{2}))\leq|t_{2}-t_{1}|$ for any
$a\leq t_{1},t_{2}\leq b$. Write
$\bar{\gamma}(t_{1})^{-1}*\bar{\gamma}(t_{2})=\sum_{j}\bar{p}_{j}$ with
$\bar{p}_{j}=\bar{p}_{j}(t_{1},t_{2})\in\bar{V}_{j}$. Since $\bar{\gamma}$ is
a horizontal curve, we have $\bar{p}_{j}=o(t_{2}-t_{1})$ for $j\geq 2$ (as
$t_{2}\rightarrow t_{1}$). If $\bar{\gamma}$ has tangent at $t_{1}$, then
$\frac{\bar{p}_{1}}{t_{2}-t_{1}}\rightarrow\bar{\gamma}^{\prime}(t_{1})$ as
$t_{2}\rightarrow t_{1}$.

For $k\geq 1$, we have

$\pi_{k\alpha+j_{0}}K(\bar{\gamma}(t_{1}),\bar{\gamma}(t_{2}))=s_{k\alpha+j_{0}}(\bar{\gamma}(t_{2}))-s_{k\alpha+j_{0}}(\bar{\gamma}(t_{1}))+[\bar{p}_{1},s_{(k-1)\alpha+j_{0}}(\bar{\gamma}(t_{1}))]+o(t_{2}-t_{1}).$

Since $d_{CC}(\bar{\gamma}(t_{1}),\bar{\gamma}(t_{2}))\leq|t_{2}-t_{1}|$, the
assumption implies that

$\frac{\pi_{k\alpha+j_{0}}K(\bar{\gamma}(t_{1}),\bar{\gamma}(t_{2}))}{t_{2}-t_{1}}\rightarrow
0$

as $t_{2}\rightarrow t_{1}$. If $\bar{\gamma}$ has tangent at $t_{1}$, then

$\frac{d}{dt}s_{k\alpha+j_{0}}(\bar{\gamma}(t))|_{t=t_{1}}=\lim_{t_{2}\rightarrow
t_{1}}\frac{s_{k\alpha+j_{0}}(\bar{\gamma}(t_{2}))-s_{k\alpha+j_{0}}(\bar{\gamma}(t_{1}))}{t_{2}-t_{1}}=[s_{(k-1)\alpha+j_{0}}(\bar{\gamma}(t_{1})),\bar{\gamma}^{\prime}(t_{1})].$

By the fundamental theorem of calculus, we have

(6)
$s_{k\alpha+j_{0}}(\bar{\gamma}(b))-s_{k\alpha+j_{0}}(\bar{\gamma}(a))=\int_{a}^{b}[s_{(k-1)\alpha+j_{0}}(\bar{\gamma}(t)),\bar{\gamma}^{\prime}(t)]dt=\int_{\bar{\gamma}}[s_{(k-1)\alpha+j_{0}}(x),\theta_{H}(x)].$

In particular, $\int_{\bar{\gamma}}[s_{(k-1)\alpha+j_{0}}(x),\theta_{H}(x)]=0$
when $\bar{\gamma}$ is a closed horizontal curve in
$\mathfrak{n}/\mathfrak{w}$ and so $s_{(k-1)\alpha+j_{0}}\in
P_{(k-1)\alpha+j_{0}}$. Furthermore, (6) also shows
$s_{k\alpha+j_{0}}=s_{(k-1)\alpha+j_{0}}^{(1)}$. By induction this shows that
$s_{j_{0}}\in\mathcal{H}_{j_{0}}$ and $s_{k\alpha+j_{0}}=s_{j_{0}}^{(k)}$ for
all $k\geq 1$.

∎

######  Proposition 6.4.

Let $s:\mathfrak{n}/\mathfrak{w}\rightarrow Z(\mathfrak{w})$ be a continuous
map satisfying $s(0)=0$ and $F:\mathfrak{n}\rightarrow\mathfrak{n}$,
$F(g)=g*s(\bar{g})$ be the associated shear map. Write $s=\sum_{j}s_{j}$,
where $s_{j}:\mathfrak{n}/\mathfrak{w}\rightarrow Z_{j}(\mathfrak{w})$ is the
$W_{j}$ component of $s$. Assume $F$ is biLipschitz.
(1) If $\alpha$ is not an integer, then $s_{j}\in E_{j}$ for $1\leq j<\alpha$
and $s_{j}\equiv 0$ for $j>\alpha$;
(2) If $\alpha$ is an integer, then $s_{j}\in\mathcal{H}_{j}$ for each $1\leq
j\leq\alpha$ and $s_{k\alpha+j}=s_{j}^{(k)}$ for all $k\geq 1$ and $1\leq
j\leq\alpha$.

###### Proof.

There is some $L>0$ such that $d(F(g_{1}),F(g_{2}))\leq L\cdot d(g_{1},g_{2})$
for any $g_{1},g_{2}\in\mathfrak{n}$. Let $g_{1},g_{2}\in\mathfrak{n}$. Then
$F(g_{1})=g_{1}*s(\bar{g}_{1})$ and $F(g_{2})=g_{2}*s(\bar{g}_{2})$. Since

$(F(g_{1}))^{-1}*F(g_{2})=(s(\bar{g}_{1}))^{-1}*g_{1}^{-1}*g_{2}*s(\bar{g}_{2}),$

we have
$K(\bar{g}_{1},\bar{g}_{2})=(g_{1}^{-1}*g_{2})^{-1}*(F(g_{1}))^{-1}*F(g_{2})$.
It follows that

$\displaystyle d(0,K(\bar{g}_{1},\bar{g}_{2}))$
$\displaystyle=d(g_{1}^{-1}*g_{2},(F(g_{1}))^{-1}*F(g_{2}))$
$\displaystyle\leq d(g_{1}^{-1}*g_{2},0)+d(0,(F(g_{1})^{-1}*F(g_{2})))$
$\displaystyle=d(g_{1},g_{2})+d(F(g_{1}),F(g_{2}))\leq(L+1)d(g_{1},g_{2}).$

Hence for any $i\geq 1$ we have

$\displaystyle|\pi_{i}(K(\bar{g}_{1},\bar{g}_{2}))|^{\frac{1}{i}}\leq
d(0,K(\bar{g}_{1},\bar{g}_{2}))$
$\displaystyle\leq(L+1)\inf_{w_{1},w_{2}\in\mathfrak{w}}d(g_{1}*w_{1},g_{2}*w_{2})$
$\displaystyle=(L+1)d(g_{1}*\mathfrak{w},g_{2}*\mathfrak{w})$
$\displaystyle\leq(L+1)Cd_{CC}(\bar{g}_{1},\bar{g}_{2})^{\frac{1}{\alpha}},$

for some $C>0$ as the distance between cosets
$d(g_{1}*\mathfrak{w},g_{2}*\mathfrak{w})$ is comparable with
$d_{CC}(\bar{g}_{1},\bar{g}_{2})^{\frac{1}{\alpha}}$. The proposition now
follows from Lemma 6.3.

∎

######  Lemma 6.5.

Given any Carnot group $G$, there exists an integer $n_{0}$ and a constant
$C>0$ with the following property: for any $g\in G$, there is a horizontal
curve $c$ from $0$ to $g$ that is a concatenation of at most $n_{0}$
horizontal line segments and such that the length of $c$ is at most $C\cdot
d(0,g)$. Here a horizontal line segment in $G$ is a path $c:[0,a]\rightarrow
G$ of the form $c(t)=g\,{\text{exp}}(tX)$, where $X$ lies in the first layer
of the Lie algebra of $G$.

###### Proof.

This follows from the proof of Proposition 2.26 in [BLD], see also Chapter 8
of [AS04] (in particular Theorem 8.1 and Proposition 8.5).

∎

Recall that $H_{1}\subset V_{\alpha}$ is a linear subspace complimentary to
$W_{\alpha}$.

######  Lemma 6.6.

There is a constant $C>0$ depending only on $N$ with the following property.
For any $b_{1},b_{2}>0$, and any $w\in Z(\mathfrak{w})$, $h\in H_{1}$
satisfying $d(0,w)\leq b_{1}^{\frac{1}{\alpha}}$ and $|h|<b_{2}$, the
inequality $d(0,(-h)*w*h)\leq
C\cdot(\max\\{b_{1},b_{2}\\})^{\frac{1}{\alpha}}$ holds.

###### Proof.

Write $w=w_{1}+\cdots+w_{m}$ with $w_{j}\in W_{j}$. The assumption $d(0,w)\leq
b_{1}^{\frac{1}{\alpha}}$ implies $|w_{j}|\leq b_{1}^{\frac{j}{\alpha}}$ for
each $j$. We calculate

$(-h)*w*h=w+\sum_{k\geq 1}\frac{(-1)^{k}}{k!}(ad(h))^{k}w.$

As $N$ is nilpotent, the above is a finite sum. We have

$\pi_{j}((-h)*w*h)=w_{j}+\sum_{k\geq 1,j-k\alpha\geq
1}\frac{(-1)^{k}}{k!}(ad(h))^{k}w_{j-k\alpha}.$

By using $|[X,Y]|\leq C_{0}\cdot|X|\cdot|Y|$, we get (with
$b=\max\\{b_{1},b_{2}\\}$)

$\huge|\frac{(-1)^{k}}{k!}(ad(h))^{k}w_{j-k\alpha}\huge|\leq\frac{C_{0}^{k}}{k!}|h|^{k}|w_{j-k\alpha}|\leq\frac{C_{0}^{k}}{k!}b_{2}^{{k}}b_{1}^{\frac{j-k\alpha}{\alpha}}\leq\frac{C_{0}^{k}}{k!}b^{\frac{j}{\alpha}}.$

From this it is clear that $|\pi_{j}((-h)*w*h)|\leq Cb^{\frac{j}{\alpha}}$ for
some $C$ depending only on $N$ and the lemma follows.

∎

######  Proposition 6.7.

Let $s_{j}\in\mathcal{H}_{j}$ be given for each integer $1\leq j\leq\alpha$.
(1) If $\alpha$ is not an integer, set $s_{j}\equiv 0$ for $j>\alpha$ and
$s=\sum_{j}s_{j}$;
(2) If $\alpha$ is an integer, set $s_{k\alpha+j}=s_{j}^{(k)}$ for all $k\geq
1$ and $s=\sum_{j}s_{j}$.
Then the shear map associated to $s$ is biLipschitz.

###### Proof.

We shall prove that the shear map $F$ associated to $s$ is Lipschitz. The same
argument shows $F^{-1}$ is also Lipschitz since $F^{-1}$ is the shear map
associated to $-s$.

By the triangle inequality,

$\displaystyle d(F(g_{1}),F(g_{2}))$
$\displaystyle=d(0,F(g_{1})^{-1}*F(g_{2}))$
$\displaystyle=d((g_{1}^{-1}*g_{2})^{-1},(g_{1}^{-1}*g_{2})^{-1}*F(g_{1})^{-1}*F(g_{2}))$
$\displaystyle\leq
d((g_{1}^{-1}*g_{2})^{-1},0)+d(0,(g_{1}^{-1}*g_{2})^{-1}*F(g_{1})^{-1}*F(g_{2}))$
$\displaystyle=d(g_{1},g_{2})+d(0,K(\bar{g}_{1},\bar{g}_{2})).$

Hence it suffices to show there is some $C>0$ such that

(7) $d(0,K(\bar{g}_{1},\bar{g}_{2}))\leq C\cdot d(g_{1},g_{2})\;\;\;\forall
g_{1},g_{2}\in\mathfrak{n}.$

(1) Assume $\alpha$ is not an integer. In this case we have
$K(\bar{g}_{1},\bar{g}_{2})=s(\bar{g}_{2})-s(\bar{g}_{1})$ and (7) follows
from the assumption.

(2) Assume $\alpha$ is an integer. We first show (7) in the case when
$\bar{g}_{1}^{-1}*\bar{g}_{2}$ lies in $\bar{V}_{1}$. So assume
$\bar{g}_{1}^{-1}*\bar{g}_{2}=t\bar{h}\in\bar{V}_{1}$ with $|\bar{h}|=1$ and
$t>0$. We have

$\displaystyle K(\bar{g}_{1},\bar{g}_{2})$
$\displaystyle=s(\bar{g}_{2})-s(\bar{g}_{1})+[-t\bar{h},-s(\bar{g}_{1})]+\sum_{k\geq
2}\frac{1}{k!}(ad(-t\bar{h}))^{k}(-s(\bar{g}_{1}))$
$\displaystyle=s(\bar{g}_{2})-s(\bar{g}_{1})+t[\bar{h},s(\bar{g}_{1})]+\sum_{k\geq
2}\frac{(-1)^{k+1}t^{k}}{k!}(ad(\bar{h}))^{k}s(\bar{g}_{1}).$

Since in our case $d(g_{1},g_{2})\geq
d_{CC}(\bar{g}_{1},\bar{g}_{2})^{\frac{1}{\alpha}}=t^{\frac{1}{\alpha}}$, it
suffice to show that there is some constant $C>0$ such that for each $j\geq
1$, the following inequality holds:

(8) $|\pi_{j}K(\bar{g}_{1},\bar{g}_{2})|^{\frac{1}{j}}\leq C\cdot
t^{\frac{1}{\alpha}}.$

When $1\leq j\leq\alpha$,
$\pi_{j}K(\bar{g}_{1},\bar{g}_{2})=s_{j}(\bar{g}_{2})-s_{j}(\bar{g}_{1})$ and
so (8) holds in this case since by assumption $s_{j}\in E_{j}$. Now let
$k_{0}\geq 1$ and suppose there is some constant $C>0$ such that
$|\pi_{k\alpha+j}K(\bar{g}_{1},\bar{g}_{2})|^{\frac{1}{k\alpha+j}}\leq C\cdot
t^{\frac{1}{\alpha}}$ holds for all $k<k_{0}$ and all $1\leq j\leq\alpha$. We
shall show a similar inequality holds for $k=k_{0}$ (with a different
constant). We have

$\displaystyle\pi_{k_{0}\alpha+j}K(\bar{g}_{1},\bar{g}_{2})$
$\displaystyle=s_{k_{0}\alpha+j}(\bar{g}_{2})-s_{k_{0}\alpha+j}(\bar{g}_{1})+t[\bar{h},s_{(k_{0}-1)\alpha+j}(\bar{g}_{1})]+\sum_{k=2}^{k_{0}}\frac{(-1)^{k+1}t^{k}}{k!}(ad(\bar{h}))^{k}(s_{(k_{0}-k)\alpha+j}(\bar{g}_{1})).$

Let $\bar{\gamma}:[0,t]\rightarrow\mathfrak{n}/\mathfrak{w}$ be the horizontal
curve given by $\bar{\gamma}(t_{1})=\bar{g}_{1}*(t_{1}\bar{h})$. We have
$\bar{\gamma}^{\prime}(t_{1})=\bar{h}$ for all $t_{1}$. By the definition of
$s_{k\alpha+j}$, we have

$\displaystyle
s_{k_{0}\alpha+j}(\bar{g}_{2})-s_{k_{0}\alpha+j}(\bar{g}_{1})+t[\bar{h},s_{(k_{0}-1)\alpha+j}(\bar{g}_{1})]$
$\displaystyle=\int_{0}^{t}[s_{(k_{0}-1)\alpha+j}(\bar{g}_{1}*(t_{1}\bar{h})),\bar{h}]dt_{1}-\int_{0}^{t}[s_{(k_{0}-1)\alpha+j}(\bar{g}_{1}),\bar{h}]dt_{1}$
$\displaystyle=\int_{0}^{t}[s_{(k_{0}-1)\alpha+j}(\bar{g}_{1}*(t_{1}\bar{h}))-s_{(k_{0}-1)\alpha+j}(\bar{g}_{1}),\bar{h}]dt_{1}$
$\displaystyle=\int_{0}^{t}[\int_{0}^{t_{1}}[s_{(k_{0}-2)\alpha+j}(\bar{g}_{1}*(t_{2}\bar{h})),\bar{h}]dt_{2},\bar{h}]dt_{1}$
$\displaystyle=\int_{0}^{t}\int_{0}^{t_{1}}[[s_{(k_{0}-2)\alpha+j}(\bar{g}_{1}*(t_{2}\bar{h})),\bar{h}],\bar{h}]dt_{2}dt_{1}$
$\displaystyle=(-1)^{2}\int_{0}^{t}\int_{0}^{t_{1}}(ad(\bar{h}))^{2}(s_{(k_{0}-2)\alpha+j}(\bar{g}_{1}*(t_{2}\bar{h}))dt_{2}dt_{1}.$

On the other hand, notice

$\displaystyle\frac{(-1)^{k+1}t^{k}}{k!}(ad(\bar{h}))^{k}(s_{(k_{0}-k)\alpha+j}(\bar{g}_{1}))=-(-1)^{k}\int_{0}^{t}\int_{0}^{t_{1}}\cdots\int_{0}^{t_{k-1}}(ad(\bar{h}))^{k}(s_{(k_{0}-k)\alpha+j}(\bar{g}_{1}))dt_{k}\cdots
dt_{1}.$

By induction we get

$\displaystyle K_{k_{0}\alpha+j}(\bar{g}_{1},\bar{g}_{2})$
$\displaystyle=(-1)^{k_{0}}\int_{0}^{t}\cdots\int_{0}^{t_{k_{0}-1}}(ad(\bar{h}))^{k_{0}}(s_{j}(\bar{g}_{1}*t_{k_{0}}\bar{h})-s_{j}(\bar{g}_{1}))dt_{k_{0}}\cdots
dt_{1}.$

There is a constant $C_{1}>0$ depending only on $\mathfrak{n}$ such that
$|[X,Y]|\leq C_{1}|X|\cdot|Y|$ for any $X,Y\in\mathfrak{n}$. Since $s_{j}$ is
$\frac{j}{\alpha}$-Holder for $1\leq j\leq\alpha$, we have

$\displaystyle|K_{k_{0}\alpha+j}(\bar{g}_{1},\bar{g}_{2})|$
$\displaystyle\leq\int_{0}^{t}\cdots\int_{0}^{t_{k_{0}-1}}|(ad(\bar{h}))^{k_{0}}(s_{j}(\bar{g}_{1}*t_{k_{0}}\bar{h})-s_{j}(\bar{g}_{1}))|dt_{k_{0}}\cdots
dt_{1}$
$\displaystyle\leq\int_{0}^{t}\cdots\int_{0}^{t_{k_{0}-1}}C_{1}^{k_{0}}|\bar{h}|^{k_{0}}|(s_{j}(\bar{g}_{1}*t_{k_{0}}\bar{h})-s_{j}(\bar{g}_{1}))|dt_{k_{0}}\cdots
dt_{1}$ $\displaystyle\leq
C_{1}^{k_{0}}\int_{0}^{t}\cdots\int_{0}^{t_{k_{0}-1}}Cd_{CC}(\bar{g}_{1}*(t_{k_{0}}\bar{h}),\bar{g}_{1})^{\frac{j}{\alpha}}dt_{k_{0}}\cdots
dt_{1}$
$\displaystyle=C_{1}^{k_{0}}C\int_{0}^{t}\cdots\int_{0}^{t_{k_{0}-1}}t_{k_{0}}^{\frac{j}{\alpha}}dt_{k_{0}}\cdots
dt_{1}$ $\displaystyle=C_{2}t^{\frac{k_{0}\alpha+j}{\alpha}}.$

Hence (8) and in turn (7) holds when
${\bar{g}_{1}}^{-1}*\bar{g}_{2}\in\bar{V}_{1}$.

Now consider the general case. Let $g,g^{\prime}\in\mathfrak{n}$. As
$\mathfrak{n}/\mathfrak{w}$ is Carnot, by Lemma 6.5, there exist
$\bar{g}={\bar{g}}_{0},{\bar{g}}_{1},\cdots,{\bar{g}}_{k}={\bar{g}}^{\prime}$
such that $k\leq n_{0}$, ${\bar{g}}_{i-1}^{-1}*{\bar{g}}_{i}\in\bar{V}_{1}$
and $\sum_{i}d_{CC}({\bar{g}}_{i-1},{\bar{g}}_{i})\leq
Cd_{CC}(\bar{g},{\bar{g}}^{\prime})$, where $C>0$ and $n_{0}$ depend only on
$\mathfrak{n}/\mathfrak{w}$. In particular we have
$d_{CC}({\bar{g}}_{i-1},{\bar{g}}_{i})\leq
Cd_{CC}(\bar{g},{\bar{g}}^{\prime})$. Let $g_{0}=g$. We inductively define
$g_{i}$ for $1\leq i\leq k$ such that
$d(g_{i},g_{i-1})=d(g_{i}*\mathfrak{w},g_{i-1}*\mathfrak{w})$. There is a
constant $C_{4}>0$ such that
$\frac{1}{C_{4}}d_{CC}(\bar{x},\bar{y})^{\frac{1}{\alpha}}\leq
d(x*\mathfrak{w},y*\mathfrak{w})\leq
C_{4}d_{CC}(\bar{x},\bar{y})^{\frac{1}{\alpha}}$ for all $x,y\in\mathfrak{n}$.
We have

$\displaystyle d(g,g_{k})$
$\displaystyle\leq\sum_{i=1}^{k}d(g_{i-1},g_{i})=\sum_{i=1}^{k}d(g_{i-1}*\mathfrak{w},g_{i}*\mathfrak{w})$
$\displaystyle\leq\sum_{i=1}^{k}C_{4}d_{CC}({\bar{g}}_{i-1},{\bar{g}}_{i})^{\frac{1}{\alpha}}\leq
C_{4}n_{0}C^{\frac{1}{\alpha}}d_{CC}({\bar{g}},{\bar{g}}^{\prime})^{\frac{1}{\alpha}}$
$\displaystyle\leq
C_{4}^{2}n_{0}C^{\frac{1}{\alpha}}d(g*\mathfrak{w},g^{\prime}*\mathfrak{w})\leq
C_{4}^{2}n_{0}C^{\frac{1}{\alpha}}d(g,g^{\prime}).$

By the triangle inequality we have
$d(g_{k},g^{\prime})\leq(1+C_{4}^{2}n_{0}C^{\frac{1}{\alpha}})d(g,g^{\prime})$.
Now by the special case, we have

$d(F(g),F(g_{k}))\leq\sum_{i}d(F(g_{i-1}),F(g_{i}))\leq\sum_{i}C_{3}d(g_{i-1},g_{i})\leq
C_{3}C_{4}^{2}n_{0}C^{\frac{1}{\alpha}}d(g,g^{\prime}).$

On the other hand it is easy to see that the restriction of $F$ on the cosets
of $W$ are isometries and so
$d(F(g_{k}),F(g^{\prime}))=d(g_{k},g^{\prime})\leq(1+C_{4}^{2}n_{0}C^{\frac{1}{\alpha}})d(g,g^{\prime})$.
Finally by the triangle inequality $d(F(g),F(g^{\prime}))\leq
d(F(g),F(g_{k}))+d(F(g_{k}),F(g^{\prime}))\leq(1+(1+C_{3})C_{4}^{2}n_{0}C^{\frac{1}{\alpha}})d(g,g^{\prime})$.

∎

##  7\. Eliminating $s_{j}$ for $j<\alpha$

The goal of this section is to prove the following result.

######  Proposition 7.1.

Let $(N,D)$ be Carnot-by-Carnot satisfying $\dim(W)\geq 2$, $\dim(N/W)\geq 2$,
and $\Gamma$ a uniform quasisimilarity group of $N$. Then there exists a
biLipschitz map $F_{0}$ such that every element in the conjugate $F_{0}\Gamma
F_{0}^{-1}$ has a compatible expression $a*Bh*Aw*As(\bar{h})$ where the
$W_{j}$ component $s_{j}$ of $s$ vanishes for every $1\leq j<\alpha$.

We shall use the approach in [DyX16] to prove Proposition 7.1. The cases
$\dim(W)=1$ and $\dim(N/W)=1$ will be considered in Sections 9 and 10
respectively.

Let $N$ and $\Gamma$ be as in Proposition 7.1. After replacing $\Gamma$ with a
biLipschitz conjugate we may assume $\Gamma$ is a fiber similarity group (see
Section 2 for the definition of fiber similarity map and fiber similarity
group). To be more precise, after applying Theorem 3.7 (and Lemma 5.1) we may
assume there are Carnot metrics $d_{CC}$ and $\bar{d}_{CC}$ on $W$ and $N/W$
respectively such that every $\gamma\in\Gamma$ induces a similarity
$\bar{\gamma}$ of $(N/W,\bar{d}_{CC})$, and there is a graded automorphism
$A_{\gamma}$ of $W$ that is also a similarity of $(W,d_{CC})$ such that
$A_{\gamma}=\gamma_{p}|_{W}$ for any $p\in N$.

Let $\text{Aut}_{I}(W,d_{CC})$ be the group of isometric graded automorphisms
of $(W,d_{CC})$, and $\text{Aut}_{c}(W,d_{CC})$ be the group of graded
automorphisms of $W$ that are compositions of a Carnot dilation and an
isometric graded automorphism. Then $\text{Aut}_{I}(W,d_{CC})$ is compact and
$\text{Aut}_{c}(W,d_{CC}))\cong\text{Aut}_{I}(W,d_{CC})\times\mathbb{R}$ is
amenable. Similarly $\text{Aut}_{c}(\mathfrak{n}/\mathfrak{w},\bar{d}_{CC})$
is amenable and so is the group of similarities
$\text{Sim}(\mathfrak{n}/\mathfrak{w},\bar{d}_{CC})=(\mathfrak{n}/\mathfrak{w})\rtimes\text{Aut}_{c}(\mathfrak{n}/\mathfrak{w},\bar{d}_{CC})$
of $(\mathfrak{n}/\mathfrak{w},\bar{d}_{CC})$. Denote
$G_{0}=\text{Aut}_{c}(W,d_{CC})\times\text{Sim}(\mathfrak{n}/\mathfrak{w},\bar{d}_{CC})$
and define a map $\Psi:\Gamma\rightarrow G_{0}$ by
$\Psi(\gamma)=(A_{\gamma},\overline{\gamma})$. It is easy to see that $\Psi$
is a homomorphism. Let $G$ be the closure of $\Psi(\Gamma)$ in $G_{0}$. We
notice that $G$ is amenable, being a closed subgroup of the amenable group
$G_{0}$.

Let $1\leq j<\alpha$ and
$E_{j}=\\{s:(\mathfrak{n}/\mathfrak{w},\bar{d}_{CC})\rightarrow
Z_{j}(\mathfrak{w})|s\;\text{is
}\;\frac{j}{\alpha}-\text{Holder},\;s(0)=0\\}$. We shall define an affine
action of $G$ on $E_{j}$, then show that this affine action has a fixed point
in $\mathcal{H}_{j}\subset E_{j}$, and finally use this fixed point to
construct a biLipschitz shear map of $N$ to conjugate $\Gamma$ into a group
with the desired property.

###  7.1. The affine action on $E_{j}$

We first define a norm on $E_{j}$, then a linear isometric action on $E_{j}$,
and finally the translational part of the affine action.

Since the compact group $\text{Aut}_{I}(W,d_{CC})$ leaves $W_{j}$ invariant,
there exists a $\text{Aut}_{I}(W,d_{CC})$-invariant inner product
$\langle,\rangle_{j}$ on $W_{j}$. Let $|\cdot|_{j}$ be the associated norm on
$W_{j}$. Notice that $E_{j}$ is a Banach space with respect to the norm

$||s||=\sup_{p\not=q}\frac{|s(p)-s(q)|_{j}}{(\bar{d}_{CC}(p,q))^{\frac{j}{\alpha}}}.$

For a similarity $f:X\rightarrow X$ of a metric space $X$, we denote by
$\lambda_{f}$ the similarity constant of $f$:
$d(f(x_{1}),f(x_{2}))=\lambda_{f}d(x_{1},x_{2})$. Let $G_{1}:=\\{(A,B)\in
G_{0}|\lambda_{B}=\lambda_{A}^{\alpha}\\}$. We first define an action of
$G_{1}$ on $E_{j}$. Let $c\in E_{j}$ and $(A,B)\in G_{1}$. We define
$\pi_{(A,B)}c:\mathfrak{n}/\mathfrak{w}\rightarrow Z_{j}(\mathfrak{w})$ to be
the function given by

$(\pi_{(A,B)}c)(\bar{h})=A^{-1}c(B(\bar{h}))-A^{-1}c(B(0)).$

Clearly $(\pi_{(A,B)}c)(0)=0$ and $\pi_{(A,B)}c$ is linear in $c$.

######  Lemma 7.2.

$\pi_{(A,B)}$ defines an action of the opposite group $G_{1}^{*}$ of $G_{1}$
on $E_{j}$ by linear isometries.

###### Proof.

Write $A=\delta_{\lambda_{A}}\circ A^{\prime}$, where $\delta_{\lambda_{A}}$
is a Carnot dilation and $A^{\prime}\in\text{Aut}_{I}(W,d_{CC})$. For $w\in
W_{j}$ we have $A(w)=\lambda_{A}^{j}A^{\prime}(w)$ and so
$|A(w_{1})-A(w_{2})|_{j}=\lambda_{A}^{j}|w_{1}-w_{2}|_{j}$ for $w_{1},w_{2}\in
W_{j}$. Now let $\bar{h}_{1},\bar{h}_{2}\in\mathfrak{n}/\mathfrak{w}$ and
$c\in E_{j}$. Then

$\displaystyle\frac{|(\pi_{(A,B)}c)(\bar{h}_{1})-(\pi_{(A,B)}c)(\bar{h}_{2})|_{j}}{\bar{d}_{CC}(\bar{h}_{1},\bar{h}_{2})^{\frac{j}{\alpha}}}$
$\displaystyle=\frac{|A^{-1}c(B(\bar{h}_{1}))-A^{-1}c(B(\bar{h}_{2}))|_{j}}{\bar{d}_{CC}(\bar{h}_{1},\bar{h}_{2})^{\frac{j}{\alpha}}}$
$\displaystyle=\frac{\lambda_{A}^{-j}|c(B(\bar{h}_{1}))-c(B(\bar{h}_{2}))|_{j}}{\bar{d}_{CC}(\bar{h}_{1},\bar{h}_{2})^{\frac{j}{\alpha}}}$
$\displaystyle=\frac{|c(B(\bar{h}_{1}))-c(B(\bar{h}_{2}))|_{j}}{(\lambda_{B}\bar{d}_{CC}(\bar{h}_{1},\bar{h}_{2}))^{\frac{j}{\alpha}}}$
$\displaystyle=\frac{|c(B(\bar{h}_{1}))-c(B(\bar{h}_{2}))|_{j}}{(\bar{d}_{CC}(B(\bar{h}_{1}),B(\bar{h}_{2})))^{\frac{j}{\alpha}}}.$

showing that $\pi_{(A,B)}c\in E_{j}$ and $||\pi_{(A,B)}c||=||c||$. We already
observed that $\pi_{(A,B)}c$ is linear in $c$. Therefore the map
$E_{j}\rightarrow E_{j}$, $c\mapsto\pi_{(A,B)}c$, is a linear isometry.

It is easy to check that
$\pi_{(A_{2},B_{2})(A_{1},B_{1})}=\pi_{(A_{1},B_{1})}\circ\pi_{(A_{2},B_{2})}$
holds for any $(A_{1},B_{1}),(A_{2},B_{2})\in G_{1}$ and so $\pi_{(A,B)}$
defines an action of the opposite group $G_{1}^{*}$ of $G_{1}$ on $E_{j}$.

∎

To obtain a linear isometric action of $G^{*}$ on $E_{j}$, we will show that
$G\subset G_{1}$.

######  Lemma 7.3.

The formula $\lambda_{\bar{\gamma}}=\lambda^{\alpha}_{A_{\gamma}}$ holds for
all $\gamma\in\Gamma$.

###### Proof.

Since both $d_{CC}$ and $d|_{W}$ ($d$ is a fixed $D$-homogeneous distance on
$N$) are homogeneous distances on $W$, there is a constant $L\geq 1$ such that
$d_{CC}(w_{1},w_{2})/L\leq d(w_{1},w_{2})\leq Ld_{CC}(w_{1},w_{2})$ for all
$w_{1},w_{2}\in W$. Similarly, as both $\bar{d}$ (recall $\bar{d}$ is the
distance on $N/W$ induced by $d$, see end of Section 3.5) and
$\bar{d}_{CC}^{\frac{1}{\alpha}}$ are $\bar{D}$-homogeneous distances on
$N/W$, there is a constant $\bar{L}\geq 1$ such that
$\bar{d}^{\frac{1}{\alpha}}_{CC}(x,y)/{\bar{L}}\leq\bar{d}(x,y)\leq\bar{L}\bar{d}^{\frac{1}{\alpha}}_{CC}(x,y)$
for any $x,y\in N/W$. After possibly replace both $L$ and $\bar{L}$ with
$\text{max}(L,\bar{L})$ we may assume $\bar{L}=L$.

Let $\gamma\in\Gamma$. Let $w_{1}\not=w_{2}\in W$. As $\gamma$ is a
$(\Lambda,C_{\gamma})$-quasi-similarity with respect to $d$ for some
$C_{\gamma}>0$, we have

$\displaystyle\lambda_{A_{\gamma}}d_{CC}(w_{1},w_{2})$
$\displaystyle=d_{CC}(\gamma_{p}(w_{1}),\gamma_{p}(w_{2}))\leq
Ld(\gamma_{p}(w_{1}),\gamma_{p}(w_{2}))$
$\displaystyle=Ld(\gamma(pw_{1}),\gamma(pw_{2}))\leq L\Lambda
C_{\gamma}d(pw_{1},pw_{2})$ $\displaystyle\leq L^{2}\Lambda
C_{\gamma}d_{CC}(w_{1},w_{2}),$

and so $\lambda_{A_{\gamma}}\leq L^{2}\Lambda C_{\gamma}$. Similarly we get
$\lambda_{A_{\gamma}}\geq\frac{C_{\gamma}}{L^{2}\Lambda}$ from

$\displaystyle\lambda_{A_{\gamma}}d_{CC}(w_{1},w_{2})$
$\displaystyle=d_{CC}(\gamma_{p}(w_{1}),\gamma_{p}(w_{2}))\geq\frac{1}{L}\cdot
d(\gamma_{p}(w_{1}),\gamma_{p}(w_{2}))$ $\displaystyle=\frac{1}{L}\cdot
d(\gamma(pw_{1}),\gamma(pw_{2}))\geq\frac{C_{\gamma}}{L\Lambda}d(pw_{1},pw_{2})$
$\displaystyle\geq\frac{C_{\gamma}}{L^{2}\Lambda}d_{CC}(w_{1},w_{2}).$

Pick $p,q\in N$ so that $pW\not=qW$ and such that $p,q$ realize the distance
between the cosets $pW$ and $qW$. Now

$\displaystyle(\lambda_{\bar{\gamma}}\bar{d}_{CC}(pW,qW))^{\frac{1}{\alpha}}$
$\displaystyle=\bar{d}^{\frac{1}{\alpha}}_{CC}(\bar{\gamma}(pW),\bar{\gamma}(qW))\leq
L\bar{d}(\bar{\gamma}(pW),\bar{\gamma}(qW))$
$\displaystyle=Ld(\gamma(p)W,\gamma(q)W)\leq Ld(\gamma(p),\gamma(q))$
$\displaystyle\leq L\Lambda C_{\gamma}d(p,q)=L\Lambda
C_{\gamma}\bar{d}(pW,qW)$ $\displaystyle\leq L^{2}\Lambda
C_{\gamma}\bar{d}^{\frac{1}{\alpha}}_{CC}(pW,qW),$

yielding $\lambda_{\bar{\gamma}}^{\frac{1}{\alpha}}\leq L^{2}\Lambda
C_{\gamma}$. Similarly by picking $p,q\in N$ so that $pW\not=qW$ and
$d(\gamma(p),\gamma(q))=d(\gamma(p)W,\gamma(q)W)$ we get
$\lambda_{\bar{\gamma}}^{\frac{1}{\alpha}}\geq\frac{C_{\gamma}}{L^{2}\Lambda}$.
Combining the above four inequalities we get
$\frac{1}{L^{4}\Lambda^{2}}\leq\frac{\lambda_{\bar{\gamma}}^{\frac{1}{\alpha}}}{\lambda_{A_{\gamma}}}\leq
L^{4}\Lambda^{2}$ for all $\gamma\in\Gamma$. Notice that for any integer
$n\geq 1$, we have $\lambda_{A_{\gamma^{n}}}=\lambda^{n}_{A_{\gamma}}$ and
$\lambda_{\bar{\gamma}^{n}}=\lambda^{n}_{\bar{\gamma}}$. The above inequality
applied to $\gamma^{n}$ yields

$\frac{1}{L^{4}\Lambda^{2}}\leq\left(\frac{\lambda_{\bar{\gamma}}^{\frac{1}{\alpha}}}{\lambda_{A_{\gamma}}}\right)^{n}\leq
L^{4}\Lambda^{2}$

for all $n\geq 1$ and so we must have
$\lambda_{\bar{\gamma}}=\lambda^{\alpha}_{A_{\gamma}}$.

∎

Lemma 7.3 implies that $\Psi(\Gamma)\subset G_{1}$. Since $G_{1}$ is a closed
subgroup of $G_{0}$, we have $G\subset G_{1}$. By restricting to $G^{*}$ the
linear isometric action of $G^{*}_{1}$ on $E_{j}$ we get a linear isometric
action of $G^{*}$ on $E_{j}$. To get an affine action on $E_{j}$ we next
define the translational part.

Define a map $b_{j}:\Gamma\rightarrow E_{j}$ by $b_{j}(\gamma)=s_{\gamma,j}$,
where $s_{\gamma,j}=\pi_{j}\circ s_{\gamma}$ and $s_{\gamma}$ is as in a
compatible expression
$\gamma(h*w)=\gamma(0)*B_{\gamma}h*A_{\gamma}w*A_{\gamma}s_{\gamma}(\bar{h})$
of $\gamma$. Recall that $\pi_{j}\circ s_{\gamma}$ is unique for $j<\alpha$
even if $\gamma$ may have more than one compatible expression. In Lemma 7.8 we
shall prove that $s_{\gamma,j}\in\mathcal{H}_{j}$; in particular,
$s_{\gamma,j}\in E_{j}$.

######  Lemma 7.4.

The equality
$b_{j}(\gamma_{2}\gamma_{1})=b_{j}(\gamma_{1})+\pi_{\Psi(\gamma_{1})}b_{j}(\gamma_{2})$
holds for any $\gamma_{1},\gamma_{2}\in\Gamma$.

###### Proof.

The proof may be tedious, but the idea is simple: it follows from two
different ways of computing $\gamma_{2}\gamma_{1}(h)$. We first notice that if
$X,Y\in\mathfrak{n}$ and $X\in\oplus_{\lambda_{j}\geq\alpha}V_{\lambda_{j}}$,
then $\pi_{j}(X*Y)=\pi_{j}(Y)$ for $1\leq j<\alpha$. We shall repeatedly use
this fact implicitly. Also recall that
$s_{\gamma,j}(\bar{h})=A^{-1}_{\gamma}\pi_{j}(\gamma(0)^{-1}*\gamma(h))$.

Let $\gamma_{1},\gamma_{2}\in\Gamma$ and $h\in H$. Write
${\gamma_{1}}(0)=h_{1}*w_{1}$ and
${\gamma_{1}}(0)*B_{\gamma_{1}}h=h_{2}*w_{2}$ with $h_{1},h_{2}\in H$ and
$w_{1},w_{2}\in\mathfrak{w}$. Then

$w^{-1}_{1}*w_{2}=[w^{-1}_{1}*(h^{-1}_{2}*h_{1})*w_{1}]*B_{\gamma_{1}}h$

and it follows that $w^{-1}_{1}*w_{2}\in\oplus_{j\geq\alpha}W_{j}$. We have
$(\gamma_{2}\circ\gamma_{1})(0)={\gamma_{2}}(0)*B_{\gamma_{2}}h_{1}*A_{\gamma_{2}}w_{1}*A_{\gamma_{2}}s_{\gamma_{2}}(\bar{h}_{1})$
and

$\displaystyle\gamma_{2}(\gamma_{1}(h))$
$\displaystyle=\gamma_{2}({\gamma_{1}}(0)*B_{\gamma_{1}}h*A_{\gamma_{1}}s_{\gamma_{1}}(\bar{h}))$
$\displaystyle=\gamma_{2}(h_{2}*w_{2}*A_{\gamma_{1}}s_{\gamma_{1}}(\bar{h}))$
$\displaystyle={\gamma_{2}}(0)*B_{\gamma_{2}}h_{2}*A_{\gamma_{2}}w_{2}*A_{\gamma_{2}}A_{\gamma_{1}}s_{\gamma_{1}}(\bar{h})*A_{\gamma_{2}}s_{\gamma_{2}}(\bar{h}_{2}).$

Now we have

$\displaystyle s_{\gamma_{2}\gamma_{1},j}(\bar{h})$
$\displaystyle=A^{-1}_{\gamma_{2}\gamma_{1}}\pi_{j}({\gamma_{2}\gamma_{1}}(0)^{-1}*(\gamma_{2}\gamma_{1})(h))$