• carnot group
• homogeneous nilpotent lie group
• quasisimilarity
• quasisymmetric maps

The purpose of this thesis is to study properties of quasisymmetric maps that preserve foliations.\\
Quasisymmetric maps are homeomorphisms between metric spaces that extend the concept of bi-Lipschitz maps. While bi-Lipschitz maps rescale the diameter of a set by no more than a multiplicative factor, quasisymmetric maps either expand or shrink the relative distance between sets.\\
More specifically, let $\eta$ be a homeomorphism of $\mathbb{R}^+$, then a $\eta$-quasisymmetric map $f : X \rightarrow Y$ is a homeomorphism such that for every three distinct point $x,y,z \in X$,
\begin{equation*}
\frac{d(f(x),f(y))}{d(f(x),f(z))} \leq \eta \Big ( \frac{d(x,y)}{d(x,z)}\Big ).
\end{equation*}

Such maps arise from the study of large scale geometry of negatively curved homogeneous manifolds. In fact, quasiisometries between negatively curved homogeneous manifolds corresponds to quasisymmetric homeomorphisms between their ideal boundaries (see \cite{EmbeddingsGromov} and \cite[Section 2]{healy2020cusped}). These boundaries have the structure of homogeneous nilpotent Lie groups, on which natural foliations are given by subgroups and their left cosets (see \cite{Heintze1974}).

Recent studies have proved that very often quasisymmetric maps on these groups, and more specifically on Carnot groups, are bi-Lipschitz (see \cite{xieFiliform}, \cite{xienonrigid}, \cite{xiereduciblecarnot} \cite{shanxie}, \cite{XIE_2017}).\\
The proof of the above-mentioned rigidity property is usually divided into two steps: the first step is to show that the quasisymmetric map preserves a certain foliation, the second step is to show that a quasisymmetric map that preserves a foliation must be bi-Lipschitz.

In this thesis, we delve into the second step exploring the general context of quasisymmetric maps on foliated quasi-metric spaces.\\
We revise some results by Le Donne and Xie (\cite{ledonnexie}), where they prove that fiber-preserving quasisymmetric maps on quasi-metric spaces endowed with a particular foliation are bi-Lipschitz.\\
More specifically, they consider the following spaces
\begin{definition}
Let $X$ be a quasi-metric space, $\alpha > 0$, and $L \geq 1$. We say that $X$ is a $(\alpha, L)$-\textit{fibered} quasi-metric space if it admits a cover $\mathcal{U}$ by closed pairwise disjoint subsets, called \textit{fibers}, with the following properties:
\begin{enumerate}
\item\label{cond1-fibre} for each fiber there exists an unbounded geodesic space $(Y,d_Y)$ such that the fiber is $L$-biLipschitz to $(Y,d_Y^{\alpha})$;
\item\label{cond2-fibre} fibers have positive distance: for every two fibers $l_1,l_2 \in \mathcal{U}$, $d(l_1,l_2)>0$;
\item\label{cond3-fibre} parallel fibers are not isolated: for every $l \in \mathcal{U}$ there exists a sequence $\{l_n\}_{n \in \mathbb{N}} \subset \mathcal{U}$ of distinct fibers parallel to $l$ such that $d_H(l_n,l) \xrightarrow[n \rightarrow \infty]{}0.$
\item\label{cond4-fibre} Non-parallel fibers diverge: for every couple of not parallel fibers \mbox{$l_1, l_2 \in \mathcal{U}$}, $d_H(l_1,l_2) = \infty$. \end{enumerate}
\end{definition}

We stress that two fibers $l_1, l_2 \subset X$ are parallel if, by definition, one has $d_H(l_1,l_2) = d(l_1,l_2)$, where $d_H$ is the Hausdorff distance.

By replacing condition \ref{cond4-fibre} of the definition of fibered quasi-metric spaces with a further requirement on the quasisymmetric map, we prove a local version of their theorem.
\begin{theorem}
Let $\alpha > 0$, $L \geq 1$, and $\eta : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ a homeomorphism.\\
Let $(X,d)$ and $(\hat{X},\hat{d})$ be $(\alpha,L)$-fibered quasi-metric spaces, satisfying only conditions \ref{cond1-fibre}, \ref{cond2-fibre}, and \ref{cond3-fibre}. Let $U \subset X$ be an open subset. Let $f : U \rightarrow f(U)$ be a $\eta$-quasisymmetric map such that:
\begin{itemize}
\item[\textbf{(F1)}] for each fiber $l \subset X$, $f$ sends all the connected components of $U \cap l$ homeomorphically onto the same fiber,
\item[\textbf{(F2)}] $f$ sends parallel fibers to parallel fibers.
\end{itemize}
Then there exists a smaller open set $V \subset U$ on which the map is a $(K,C)$-quasisimilarity.
\end{theorem}
Thus, the theorem by Le Donne and Xie is a corollary of our theorem.
Eventually, we prove that homogeneous Lie groups are fibered metric spaces with respect any homogeneous quasi-metric. Thus, our Theorem can be applied to such groups.