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% Author Package file for use with AMS-LaTeX 1.2
\controldates{26-SEP-2001,26-SEP-2001,26-SEP-2001,26-SEP-2001}
 
\documentclass{era-l}
\issueinfo{7}{11}{}{2001}
\dateposted{September 28, 2001}
\pagespan{79}{86}
\PII{S 1079-6762(01)00097-X}
\usepackage{euscript}

\copyrightinfo{2001}{American Mathematical Society}



\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{statement}[lemma]{Statement}
\newtheorem{corollary}[lemma]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[lemma]{Definition}

\theoremstyle{remark}
\newtheorem{notation}[lemma]{Notation}

\begin{document}
\title[On pairs of metrics]{On pairs of metrics invariant under a 
cocompact action of a group}
\author{S. A. Krat}
\address{Department of Mathematics, The Pennsylvania State University, 
University Park, PA 16802}
\email{krat@math.psu.edu}

\subjclass[2000]{Primary 51K05; Secondary 53C99}
\keywords{Metric space, group action}
\date{February 16, 2001}
\commby{Richard Schoen}

\begin{abstract}
Consider two intrinsic metrics invariant under the same cocompact
action of an abelian group. Assume that the ratio of the distances 
tends to one as the distances grow to infinity. Then it is known (due 
to D. Burago) that the difference between the metric functions is 
uniformly bounded.

We will prove an analog of this result for hyperbolic groups, as well 
as a partial generalization of this result for the Heisenberg group: a 
word metric on the Heisenberg group lies within bounded GH distance 
from its asymptotic cone.
\end{abstract}
\maketitle
\section*{Introduction}
Let $M$ be a set provided with two interior metrics $d_1$ and $d_2$.
Assume that a group $G$ acts cocompactly on $M$ by isometries
with respect to  both  metrics and 
\begin{equation*}
 \lim_{d_2(x,y)\to\infty} \frac{d_1(x,y)}{d_2(x,y)}=1.
\end{equation*}
Due to a  result of 
D.~Burago~\cite{bur}, if the group $G =\mathbb Z^n$, then there is a 
constant $C$ such that $|d_1(x,y)-d_2(x,y)|\le C$. 
This fact means that all metrics on $M$ diverge linearly or stay
within a finite distance from each other. 
Burago raised the question for which groups the same statement could 
be  true. He suggested two different directions.    
The first is the case of semi-hyperbolic groups, i.e.,
groups of isometries of a space whose curvature is bounded from above by 
$0$.
The other one is the case of nilpotent groups and first of all, the 
Heisenberg group. Our goal is to answer some of these questions.

The first part of  the paper is devoted to the case of hyperbolic groups.
To the best of our knowledge this result has never been published, even
though many experts probably believe in it.

In some cases the fact that two metrics cannot diverge more slowly than
linearly could be described as the finiteness of the Gromov-Hausdorff
distance between the group with induced metric and its asymptotic cone.
In the case of the abelian group $\mathbb Z^n$ the asymptotic cone is 
$\mathbb R^n$
and it lies  within a finite  Gromov-Hausdorff distance from $\mathbb 
Z^n$.
We will describe one more case  when this is true, and for the same reason, 
in the second  part of this paper. In the case of hyperbolic groups 
the result does not depend on the distance to the 
asymptotic cone because the asymptotic cone  is not the Gromov-Hausdorff 
limit of the group with corresponding metrics.
  
The second part  
reviews the case of the simplest nontrivial infinite
discrete nilpotent group, namely the discrete 
Heisenberg group $\Gamma$, where we get a partial result (for word 
metrics). 
The proof of the finiteness of the Gromov-Hausdorff distance
between $\Gamma$ and its asymptotic cone was published in~\cite{krat}.
Here we will give the main idea of the proof and will show how
 the result follows from this fact.

The next natural case which generalizes the cases of abelian and 
hyperbolic  
groups is the case of semi-hyperbolic groups.
It is not known yet whether the statement is true or false for it.

\section{Hyperbolic group}
Let a hyperbolic group $G$ act cocompactly on a
$\delta$-hyperbolic metric space $M$ by isometries with respect to two 
interior metrics $d_1$ and $d_2$.
\begin{theorem}
\label{thy1}
Assume that 
\begin{equation*}
\lim_{\max(d_1(x,y),d_2(x,y))\to\infty}\frac{d_1(x,y)}{d_2(x,y)}=1;
\end{equation*} 
then there is a constant $C$ such that $|d_1(x,y)-d_2(x,y)|\le C $ for 
each $x,y\in M$.
\end{theorem}
 
\begin{notation}
We will denote the diameter of the compact set $M/G$ by $D$ and the 
difference
$d_1(x,y)-d_2(x,y)=\Delta(x,y)$.
In this paper by a geodesic we will mean the shortest path and
consider only geodesics with natural parameterization.
\end{notation}



  
The next lemma  shows  that the geodesic in metric $d_2$ lies in the 
constant neighborhood from 
the geodesic in metric $d_1$ with the same endpoints, where the 
constant 
does not depend on the choice of geodesics or their endpoints.
This fact follows from the Morse lemma since under the conditions of the 
theorem
any geodesic in metric $d_2$
is a quasi-isometric map in the metric $d_1$.


\begin{lemma}
\label{lhy1}
Under the conditions of the theorem 
there exists a constant $c_1$ 
such that for 
each geodesic $\gamma_2:[0,l_2]\to (M,d_2)$ with endpoints $x$ and $y$ 
there is 
a geodesic $\gamma_1:[0,l_1]\to (M,d_1)$, with endpoints $x$ and $y$, 
such that 
$\gamma_2$ lies in the $c_1$-neighborhood (in the metric $d_1$) of $\gamma_1$.
\end{lemma}



\begin{corollary}
\label{chy1} 
Since the conditions of Lemma~\ref{lhy1} are symmetric
in $d_1$ and $d_2$, 
there exists a constant $c_2$ 
such that for 
each geodesic $\gamma_1:[0,l_1]\to (M,d_1)$ 
with endpoints $x$ and $y$ there is 
a geodesic $\gamma_2:[0,l_2]\to (M,d_2)$, 
with endpoints $x$ and $y$, such that 
$\gamma_1$ lies in the $c_2$-neighborhood (in the metric $d_2$) of $\gamma_2$.
\end{corollary}

From Corollary~\ref{chy1} it follows that in the $c_2$-neighborhood (in the
metric $d_2$) of
any point on the geodesic
$\gamma_1$   there is a point of the geodesic $\gamma_2$. 
Using  triangle inequalities for these two points and the endpoints of 
the
geodesics $\gamma_1$ and $\gamma_2$ one can  
 derive the quasi-additivity property of the function
$\Delta$.


\begin{lemma}
\label{l3}
If M is as described in the theorem and $\gamma_1:[0,l_1] \to M$ 
is geodesic in the metric
$d_1$, then for each $t_1\in [0,l_1]$,
\begin{equation*}
|\Delta(\gamma_1(0),\gamma_1(l_1))-\Delta(\gamma_1(0),\gamma_1(t_1))-
\Delta(\gamma_1(t_1),\gamma_1(l_1))|
\le 2c_2.
\end{equation*}
\end{lemma}

Now let us reformulate the conditions of the theorem in a more 
convenient form.

\begin{lemma}
\label{l4}
Under the conditions of the theorem 
\begin{equation*}
 \lim_{d_1(x,y)\to\infty}\frac{\Delta(x,y)}{d_1(x,y)} =0.
\end{equation*}
\end{lemma}

Since the group $G$ acts cocompactly on $M$, it is enough to
consider the points of one orbit under the action of $G$ and measure
the distances between them. We introduce a notation. 

\begin{notation}
\label{nhy2}
We choose a point $x_0\in M$, and let $G(x_0)$ be the orbit of this point 
under 
the action of the group $G$. For each $g\in G$, we will denote by $\gamma_g$ 
the geodesic
in the metric $d_1$
with endpoints $x_0$ and $g(x_0)$, and by $\Delta(g)$, the quantity 
$\Delta(x_0,g(x_0))$.
It is easy to see that
 $\Delta(g)=\Delta(g^{-1})$ for each $ g$ in $G$.
\end{notation}

Reasoning by contradiction we need to show that if the
function $\Delta(x,y)$ is unbounded, then 
\begin{equation*}
 \lim_{d_1(x,y)\to\infty}\frac{\Delta(x,y)}{d_1(x,y)} \ne 0.
\end{equation*} 

The $\delta$-hyperbolicity of the space $M$ means that if two geodesics
with the same endpoint diverge relatively fast, then the distance 
between
the other endpoints of these geodesics is almost the sum of the 
lengths of 
these geodesics.
Now we need to define what we mean by that
 two geodesics $\gamma_{g_1}$ and $\gamma_{g_2}$ diverge relatively
fast. The following function
will show the moment after which 
the distance from each point on the geodesic $\gamma_{g_1}$
to geodesic $\gamma_{g_2}$ is greater than a certain $\epsilon$. 

\begin{definition}
Let us define the function \begin{equation*}
f(g_1,g_2,\epsilon)=
\max\{t \mid d_1(\gamma_{g_1}(t),\gamma_{g_2})
\le\epsilon\},
\end{equation*} where $g_1,g_2\in G$ and 
$\epsilon\in\mathbb R$. Here by $d_1(\gamma_{g_1}(t),\gamma_{g_2})$
we mean the distance from the point $\gamma_{g_1}(t)$ to the set 
$\gamma_{g_2}$.
\end{definition}

The function $f$ is not symmetric under the interchange of $g_1$ and $g_2$.
But from the triangle inequalities it follows that the difference of 
values
$f(g_1,g_2,\epsilon)$ and $f(g_2,g_1,\epsilon)$ is small.


\begin{lemma} 
\label{lhy5}
If $f$ is a function described above, then $|f(g_1,g_2,\epsilon)
-f(g_2,g_1,\epsilon)|\le\epsilon$.
\end{lemma}


The function $f(g_1,g_2,\epsilon)$ shows us the last time when the geodesic
$\gamma_{g_1}$ came to the distance $\epsilon$ from the geodesic 
$\gamma_{g_2}$. But there remains the question whether the geodesic 
$\gamma_{g_1}$
stayed near the geodesic $\gamma_{g_2}$ before that?
The answer is yes: it stays within the distance $\epsilon+\delta$.
This fact follows from the $\delta$-hyperbolicity of $M$.

\begin{lemma}
\label{lhy6}
Let $g_1,g_2\in G$.
If $t\le f(g_1,g_2,\epsilon)$, then $d_1(\gamma_{g_1}(t),\gamma_{g_2})
\le \epsilon+\delta$.
\end{lemma}

The next lemma is some sort of analog of behavior of the angles in 
$\mathbb R^3$.
If the angle between two rays with the same starting point in $\mathbb 
R^3$
is $2\epsilon$, then each ray with the same starting point forms the
angle at least  $\epsilon$ with one of them. The 
angle between two rays in $\mathbb R^3$ shows how fast these
rays diverge.

In our case we can say that if two geodesics diverge fast to the 
distance 
$2\epsilon+2\delta$, then each geodesic with the same starting point 
quickly moves $\epsilon$ units away from one of them.
We cannot get $2\epsilon$ instead of $2\epsilon+2\delta$, as in the 
case of rays,
because the geodesics in our case can diverge and then converge again.
But the amount of this divergence is bounded by $\epsilon+\delta$ 
in view of Lemma~\ref{lhy6}. 

\begin{lemma}
\label{lhy7}
If $g_1,g_2\in G$,  $C\in \mathbb R$ and $ f(g_1,g_2,2\epsilon+2\delta)
\le C$, then
for each $g_3\in G$ either 
$f(g_3,g_1,\delta)\le C+\epsilon$ or $f(g_3,g_2,\delta)
\le C+2\delta+\epsilon$.
\end{lemma}


Now we assume that it is possible to find two 
elements $h_1$ and $h_2$ 
of the group $G$ such that the two
geodesics connecting $x_0$ with $h_1(x_0)$ and $h_2(x_0)$ 
diverge relatively fast
and  the values $\Delta(x_0,h_1(x_0))$ and 
$\Delta(x_0,h_2(x_0))$ are relatively large.
 The proof of the 
existence of these elements is based on the almost additivity of the 
function
$\Delta$ and we will give it later. 

Using the elements $h_1,   \ h_2, \ h_1^{-1}, \ h_2^{-1}$      
we will construct the sequence of elements $\{g_i\}$ such
that  the geodesics connecting 
$x_0$ with $g_i(x_0)$ and $g_{i+1}(x_0)$
also diverge relatively fast and  the $\Delta(g_i(x_0))$ are relatively 
big.
Then we will construct the piece wise geodesic path which starts at 
a point $x$
and then goes  consecutively through all the
points $x_i=g_i\circ\cdots\circ g_1(x_0)$.
It consists of the 
isometric images of geodesics connecting $x_0$ with $h_1(x_0)$ and 
$h_2(x_0)$.
From Lemma~\ref{lhy7} it follows that the geodesics connecting $x_i$
with $x_0$ and $x_{i+1}$ diverge relatively fast. 

\begin{lemma}
\label{lhy8}
If there exist  $h_1,h_2\in G$ such that 
\begin{equation*}
\min(\Delta(h_1),\Delta(h_2))\ge 
4f(h_1,h_2,6\delta)+20\delta,
\end{equation*}
then there exists a sequence $g_1,g_2,g_3,\ldots$ of elements of the group $G$ 
such that \begin{equation*}
\Delta(g_i)\ge 
4f((g_1\circ\cdots\circ g_{i-1})^{-1},g_i,2\delta)+4\delta.
\end{equation*}
 \end{lemma}

Now that we have constructed the sequence $\{g_i\}$, 
we are going to prove that
its existence contradicts the conditions of the theorem.
It follows from the $\delta$-hyperbolicity of the space that the geodesic
connecting $x_0$ and $x_i$ goes near all the points $x_n$ where $nD_0$ then $ d_1(x,y)>\frac{1}{2}\Delta(x,y)$ and 
$d_2(x,y)>\frac{1}{2}\Delta(x,y)$. 
\end{lemma}

\begin{notation}
If the conditions of the theorem hold, then the set 
\begin{equation*}
\{d_2(x,y) \mid d_1(x,y)\le\epsilon\}
\end{equation*} is bounded. We use the notation
$\max_{d_1(x,y)\le\epsilon}(d_2(x,y))=K_\epsilon$.
\end{notation}

To find the elements $h_1$ and $h_2$ we 
choose an element $g\in G$ with large $\Delta(g)$.
By Lemma~\ref{lhy10} this means that the length of 
$\gamma_g$ is also large.
Then we choose a point $y$ on the orbit of $x_0$ near the point on 
$\gamma_g$ where the function 
$\Delta$ becomes $\frac{1}{2}\Delta(g)$, which is still a large number.
And then the geodesics connecting $y$ with $x_0$ and $g(x_0)$ pass
near the geodesic $\gamma_g$ in different directions. Then they 
diverge 
fast comparative  to $\Delta(y,x_0)$ and $\Delta(y,g(x_0))$ which are close
to $\frac{1}{2} \Delta(g)$.

\begin{lemma}
Under the conditions of the theorem,
if for each $c\in\mathbb R$ there exists $g_c\in G$ such that 
$\Delta(g_c)>c$, 
then there are  two elements $g_1$ and $g_2$ such that
$\min(\Delta(g_1),\Delta(g_2))\ge 4f(g_1,g_2,6\delta)+20\delta$.
\end{lemma}

This lemma finishes the proof of Theorem~\ref{thy1}.
 
\section{The Heisenberg group}
Let the discrete Heisenberg group $\Gamma$ act cocompactly on a space $M$
with respect to two metrics $d_1$ and $d_2$. Choose a point $x_0\in M$ and
define metrics $d_1'$ and $d_2'$ on $\Gamma$ by the formulas 
\begin{equation*}
d_1'(\gamma_1,\gamma_2)=d_1(\gamma_1(x_0),\gamma_2(x_0)),
\end{equation*}
 \begin{equation*}
d_2'(\gamma_1,\gamma_2)=d_2(\gamma_1(x_0),\gamma_2(x_0)),
\end{equation*}
for $\gamma_1,\gamma_2\in\Gamma$.
The main result of this part of 
this work is the following theorem:


\begin{theorem}
 Assume that
\begin{equation*}
\lim_{\max(d_1(x,y),d_2(x,y))\to\infty}\frac{d_1(x,y)}{d_2(x,y)}=1
\end{equation*}
and $d_1'$ and $d_2'$ are word metrics on $\Gamma$, then there is a 
constant 
$C$ such that $|d_1(x,y)-d_2(x,y)|\le C$ for each $x,y\in M$.
\end{theorem}


 The proof of this theorem is based on the following statement from
\cite{krat}.


 \begin{statement}
 \label{osnovth} For each word metric on the Heisenberg group $\Gamma$,
the Gromov-Hausdorff distance between $\Gamma$ and its asymptotic cone
$Con_\infty\Gamma$ is finite.
 \end{statement}



\subsection{Preliminaries}

In this section,  we give a description
of the structure of the asymptotic cone of the Heisenberg group based
on the results in~\cite{p}.

The definition of the asymptotic cone 
and Gromov-Hausdorff distance can be found in~\cite{gr3}.
Let us now describe the structure of the asymptotic cone of the 
Heisenberg group. To do this we need to introduce the notion of  
expansion. 

  \begin{definition} {\it The expansion}  of the group 
$ \Gamma $ with coefficient
  $k$ is a map $ { \delta_k\!: \Gamma \to
\Gamma}$, $ {\delta_k(\gamma)= (kx,ky,k^2z)}$, where $ \gamma=(x,y,z), $
 $k\in \mathbb Z$.
 \end{definition}

 The expansions $\delta_k$ of the group $\Gamma$ play the same role as 
homotheties in commutative groups. 
 Let
 $\|\cdot\|:\Gamma\to\mathbb R$ be a norm on the group $\Gamma$,
i.e., a real function that satisfies the properties:
\begin{itemize}
\item $\|\gamma\|\ge 0$ for each $\gamma\in \Gamma$; $\|\gamma\|=0$ if 
and 
only if $\gamma=1$;
\item
$\|\gamma\|=\|\gamma^{-1}\|$;
\item
$\|\gamma_1\gamma_2\|\le\|\gamma_1\|\|\gamma_2\|$.
\end{itemize}

With each norm on $\Gamma$, one can associate a left-invariant metric
 on $\Gamma$
 \begin{equation*}
 |\gamma_1,\gamma_2 |=\|\gamma_1^{-1}*\gamma_2 \|. 
\end{equation*}
We recall  the definition of an interior metric on a discrete group
given by Pansu~\cite{p}.


 \begin{definition}
The left-invariant metric associated with the norm
$\|\cdot\|$ on the discrete group $G$
 is called {\it interior} if for each
 $  \varepsilon>0 $ there exists $ p>0 $ such that
 $ \forall \gamma \in G, \gamma $ can be represented as
 ${\gamma=\gamma_1*\cdots*\gamma_n}$.
 Here 
 $    \| \gamma_i \| \le p$
 and
 $    \| \gamma_1 \|+ \cdots+\| \gamma_n \| \le (1+\varepsilon)\| \gamma \|$.
 \end{definition}


From Pansu's result~\cite{p}, it follows that if the left-invariant 
metric associated with the given norm $\|\cdot\|$ on the Heisenberg 
group $\Gamma$
is interior, then the asymptotic cone of the Heisenberg group with this 
metric is the space $(H,\|\cdot\|_\infty)$, where
$H$ is a continuous Heisenberg group. 
This is just the group of all upper 
triangular real $3\times3$ matrices with $1$'s on the main diagonal. 
The left-invariant metric $d_\infty$ associated with the norm 
$\|\cdot\|_\infty$ is the Carnot-Carath\'eodory-Finsler metric given by the 
pair
$(L_0,F_0)$; here $L_0$ is the two-dimensional subspace of the Lie algebra of 
the Lie group $H$, spanned by the vectors $(1,0,0)$ and $(0,1,0)$, and 
$F_0$ is the Minkowski metric on $L_0$.    Moreover, for elements
 $\gamma\in\Gamma\subset H $ the  limit $\lim_{k\to\infty}
    \frac{\| \delta_k(\gamma) \|}{k} $  exists and the following equality holds:
 \begin{equation*}
    \| \gamma \|_\infty=\lim_{k\to\infty}
    \frac{\| \delta_k(\gamma) \|}{k}.
\end{equation*}
   The number $\| \gamma \|_\infty$  is called 
 {\it the asymptotic norm of the element
 $\gamma\in\Gamma $}.



  

  Later, we will consider only the word metrics on the Heisenberg group 
$\Gamma$. Each such  metric is associated with a finite generating set 
$E\subset \Gamma$, which we will call  {\it an alphabet.} Each element
 $\gamma\in\Gamma$ can be represented as $\gamma=a_1*\cdots*a_k, \
 a_i\in E.$  The smallest such  $k$ is called the {\it length} of the 
element $\gamma, \
 \|\gamma\|=k.$  
It is obvious that the length is a norm. The 
corresponding left-invariant metric on 
$ \Gamma$ is called {\it the word metric.}
It is easy to see that the  word metric is an interior metric on the 
Heisenberg group $\Gamma$.

 Hereinafter, we identify $E$ with the subset of our lattice which  in 
turn is embedded in $H.$ Thus, we consider $E$ to be a subset of $ H.$ 
Let us also assume that $E$ is symmetric,
 i.e., $E=E^{-1}$. In particular, $|E|=2n$ is an even number.
A finite sequence of vectors of the alphabet is called a
{\it word}.
The number of  vectors in a word is called  the {\it length} of the 
word.

 Let us reformulate Theorem~\ref{osnovth}. 
 In order to prove the fact that
\begin{equation*}
 {\bigl|(\Gamma,\|\cdot\|);({H},\|\cdot\|_\infty)\bigr|}_H\le Const,
\end{equation*}
 where $\|\cdot\|$ is a word norm on the Heisenberg group $\Gamma$, it 
is sufficient to compare two different norms 
 $\|\cdot\|$ and $\|\cdot\|_\infty$ on $\Gamma$.
 If we prove that
 $\bigl|\|\gamma\|-\|\gamma\|_\infty\bigr|\le Const$ for each
 $\gamma\in\Gamma$ and take an embedding $\Gamma\to H$,
we will get the required statement. In other words, to prove 
Theorem~\ref{osnovth}, it is sufficient to check  two inequalities:
 \begin{eqnarray}
 \|\gamma\|_\infty\le\|\gamma\|+Const,\label{os1} \\
 \|\gamma\|\le\|\gamma\|_\infty +Const.\label{os2}
 \end{eqnarray}

Let us now turn back to the metric spaces $(M,d_1)$ and $(M,d_2)$.
Recall that the Heisenberg group $\Gamma$ acts cocompactly by 
isometries on 
both of them. We choose a point $x_0\in M$.
This allows us to introduce two norms on the group $\Gamma$.
Let $\gamma$ be an element of the group $\Gamma$. Then 
$\|\gamma\|_1 =d_1 (x_0,\gamma(x_0))$ and 
$\|\gamma\|_2 =d_2 (x_0,\gamma(x_0))$. 
 These norms provide  asymptotic norms 
$\|\cdot\|_{\infty 1}$ and $\|\cdot\|_{\infty 2}$ on the group $\Gamma$.
\begin{lemma}
Let $M$ be a set provided with two metrics $d_1$ and $d_2$ such that
\begin{equation*}
 \lim_{\max(d_1(x,y),d_2 (x,y))\to\infty}\frac{d_1(x,y)}{d_2(x,y)} =1.
\end{equation*}
Let the Heisenberg group $\Gamma$ act by isometries with respect to 
both metrics. Then $\|\cdot\|_{\infty 1}=\|\cdot\|_{\infty 2}.$
\end{lemma}
\begin{proof}
Indeed, for each $\gamma\in \Gamma$ we have
\begin{equation*}
 \|\gamma\|_{\infty 1}=\lim_{k\to\infty}\frac{\|\delta_k(\gamma)\|_1}
{k}= \lim_{k\to\infty}\frac{\|\delta_k(\gamma)\|_2}{k}
\frac{\|\delta_k(\gamma)\|_1}{\|\delta_k(\gamma)\|_2} =
\lim_{k\to\infty}\frac{\|\delta_k(\gamma)\|_2}{k}= \|\gamma\|_{\infty 2}
\end{equation*} 
since $d_2(x_0,\delta_k(\gamma)(x_0))\to\infty$ as $k\to\infty$.
\end{proof}

Using 
(\ref{os1}) and (\ref{os2}) 
and the previous lemma we get that 
\begin{equation*}
 |\|\gamma\|_1-\|\gamma\|_{\infty 1}| \le Const_1,
\end{equation*}
\begin{equation*}
 |\|\gamma\|_2-\|\gamma\|_{\infty 2}| \le Const_2,
\end{equation*} and so
\begin{equation*}
 |d_1(x_0,\gamma(x_0))-d_2(x_0,\gamma(x_0))| \le Const_1+Const_2.
\end{equation*} 

Then it is enough to prove (\ref{os1}) and (\ref{os2}). 


 \section{Sketch of the proof of inequalities (\ref{os1}) and 
(\ref{os2})}

In this section we describe the main idea of the  proof of 
inequalities
(\ref{os1}) and
 (\ref{os2}). The exact proof can be found in \cite{krat}.
We start with a proof of (\ref{os1}) and (\ref{os2}) for word 
metrics
of a special kind, which we now define.
 \begin{definition}
If the third coordinates of all vectors of the alphabet $E$ are equal to
$0$, then the alphabet  is called 
{\it horizontal} and the corresponding word metric is  called a {\it 
horizontal word metric.}
 \end{definition}
 The inequality $\|\gamma\|_\infty\le\|\gamma\| $
 in the case of a horizontal word metric is an easy calculation.
 Indeed let the left-invariant metric on the Heisenberg group
$\Gamma$ associated with the norm $\|\cdot\|$ be a horizontal word metric.
Assume that
 $\gamma$ can be represented as
 $
 \gamma=a_1*\cdots*a_N;
 $
 then \begin{equation*}
\delta_k(\gamma)=\delta_k(a_1)*\cdots*\delta_k(a_N)=
 {a_1}^k*\cdots*{a_N}^k. 
\end{equation*} 

 Therefore,
 $\  \|\delta_k(\gamma)\|\le kN=k\|\gamma\| \ $ and
 \begin{equation*}
   \|\gamma\|_\infty=
   \lim\limits_{k\to\infty}\frac{1}{k}\|\delta_k(\gamma)\|\le
   N=\|\gamma\|.
\end{equation*}

The proof of the other inequality is based on the following property of
the Heisenberg group.
Let $\gamma=(x,y,z)$ be an  element of $\Gamma$. Assume that
$\gamma=a_1*\ldots *a_N$, where $a_i$ are vectors of the horizontal 
alphabet
$E$. Start at the origin and draw a closed
polygonal line consisting of the vectors $a_i$ which lie in 
$Oxy$ and the segment connecting the point $(x,y)$ with the origin.
Then $|z|$ is equal to the oriented
area enclosed in this  polygonal line.
This property follows from the formula for multiplication in the Heisenberg 
group:
$(x_1,y_1,0)*(x_2,y_2,0)=(x_1+x_2,y_1+y_2,\frac{1}{2}(x_1y_2-y_1x_2))$.
The number $\frac{1}{2}(x_1y_2-y_1x_2)$ is equal to the area of the 
triangle
with vertices $(x_1,y_1)$, $(x_2,y_2)$ and the origin.

 This allows us to calculate 
 the norm of an arbitrary element $\gamma=
 (x,y,z)$ in the horizontal word metric as  the number of elements in 
the smallest collection of vectors of the alphabet  
on the plane $Oxy$ which bounds 
the area $z$ and whose sum equals the vector $(x,y)$.
This method of norm computation gives us an opportunity to reduce the 
proof of the inequalities (\ref{os1}) and (\ref{os2}) to the comparison 
of 
isoperimetric curves on the plane. 

Indeed, consider the Minkowski metric on the plane $Oxy$, where the unit 
ball is
the convex hull of the vectors of the horizontal alphabet $E$. 
The number of the vectors in some word is equal to the length
of the corresponding polygonal line in this metric. Moreover, this 
metric, 
as follows from~\cite{ber}, is 
exactly the  metric $F_0$ that gives rise to the 
Carnot-Carath\'eodory-Finsler metric on $Con_\infty\Gamma$.
So $\|\gamma\|_\infty$ is equal to the length 
of a certain  path in the metric $F_0$.  This  path
connects the points $(0,0)$ and $ (x,y)$. It is also the shortest path that
together with the segment from $(x,y)$ to $(0,0)$ bounds the area $z$. 
As follows from~\cite[Chapter 5, Theorem 22.4]{l},
this path is a polygonal line. 
The norm $\|\gamma\|$ is equal to the length of a certain polygonal line 
in the metric $F_0$. It connects the
points $(0,0)$ and $(x,y)$. It is also the shortest
polygonal line such that all segments in it  are integer and it 
bounds the area $z$.
These two polygonal lines are almost the same and the difference of 
their 
lengths is bounded by a constant. For  details see~\cite{krat}.

To prove inequalities (\ref{os1}) and (\ref{os2}) for the case of 
general 
word metric we change the general alphabet 
$E=\{a_1=(x_1,y_1,z_1),\dots,
a_{2n}=(x_{2n},y_{2n},z_{2n})\} $ to the horizontal alphabet $E_0=
\{(x_1,y_1,0),
\dots,(x_{2n},y_{2n},0)\}$. Then we mention that the 
corresponding word 
metrics differ by a constant. To see this we check that 
the polygonal line (of projections of the alphabet vectors on $Oxy$)
corresponding to some almost 
the shortest word, which gives us some element $\gamma=(x,y,z)$,
consists of at most $2n+2$  segments. That means that the same vectors 
come 
together in the shortest word. Then we  consider each segment of the  
polygonal line. It consists of $r_i$ projections on $Oxy$ of vectors 
$a_i$. 
Change it for three
segments that finish  a parallelogram whose area is proportional
to $z_i$ and $r_i$. We will choose this parallelogram in such a way 
that
the area  bounded by the new curve is $z$. Then  the length of the new 
polygonal line is almost the norm of $\gamma$ in a horizontal word 
metric with 
alphabet $E_0$. And it differs from the length of the initial polygonal line
by a constant, depending only on the alphabet.
     
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