## Thesis etd-10232018-233050 |

Thesis type

Tesi di dottorato di ricerca

Author

KUZMIN, KIRILL

URN

etd-10232018-233050

Thesis title

Quasi-Isometric Rigidity for Universal Covers of Manifolds with a Geometric Decomposition

Academic discipline

MAT/03

Course of study

MATEMATICA

Supervisors

**tutor**Prof. Frigerio, Roberto

Keywords

- Geometric Group Theory
- Quasi-Isometric Rigidity
- Cusp-Decomposable Manifolds
- Non-Positively Curved Manifolds

Graduation session start date

07/11/2018

Availability

Full

Summary

This work is a drop in the stream of research originated by the ideas of Gromov, who pointed the attention on large scale properties of metric spaces and other mathematical objects that can be related to them, in particular finitely presented groups.

The most studied morphisms in this settings are quasi-isometric embeddings, i.e. maps $f$ between metric spaces $(E_1,d_1)$ and $(E_2,d_2)$ for which there are constants $K \geq 1$ and $c \geq 0$ such that the inequality

\[ K^{-1}d_1(p,q) - c \leq d_2 (f (p), f (q)) \leq Kd_1(p,q) + c \]

holds for every $p$, $q$ in $E$.

A quasi-isometric embedding is a quasi-isometry if there exists a quasi-isometric embedding $g \colon E_2 \to E_1$ such that the compositions $f \circ g$ and $g \circ f$ are at a bounded distance from the identity maps of $E_2$ and $E_1$ respectively.

If a group $G$ is generated by a finite set $S$, one can define $d (x,y)$ for $x$, $y$ in $G$ to be the minimal length of a string of elements of $S$ and inverses of elements of $S$ that represents $x^{-1}y$. This turns out to be a distance in $G$, invariant by left multiplication, and well defined up to bi-Lipschitz equivalence, hence quasi-isometry, if one changes the generating set. This is the starting point of Geometric Group Theory. One of the questions it tries to answer is whether certain algebraic properties of groups, or geometric properties of spaces on which these groups act nicely, are preserved under quasi-isometry.

A group is quasi-isometric to a finite index subgroup, and to any of its extensions by a finite group. These two steps generate the relation of virtual isomorphism between groups, and it follows that virtually isomorphic groups cannot be distinguished by quasi-isometries.

We can then define a class of groups to be \emph{quasi-isometrically rigid} if any group quasi-isometric to a group of the class is virtually isomorphic to a possibly different group in the class. We also say that a group is \emph{quasi-isometrically rigid} if any group quasi-isometric to it is virtually isomorphic to it.

In this thesis we address in particular the question of quasi-isometric rigidity for families of fundamental groups of certain manifolds having a canonical geometric decomposition. This decomposition comes either from the construction of the manifold, or from some operation performed on the manifold.

An example of the first case are cusp-decomposable manifolds defined by Nguy\^en-Phan. They are constructed by gluing along affine homeomorphisms of the boundary manifolds obtained by cutting away cusps of complete finite volume negatively curved locally symmetric manifolds. The precise definition will be given in Section 3. For this class of manifolds I present original results, the main of which are the following:

\begin{teo}

Let $\Gamma$ be a group quasi-isometric to the fundamental group of a cusp-decomposable manifold $M$. Then $\Gamma$ is virtually isomorphic to the fundamental group of a cusp-decomposable orbifold.

\end{teo}

\begin{teo}

The class of orbifold fundamental groups of cusp-decomposable orbifolds is quasi-isometrically rigid, i.e. every group quasi-isometric to an orbifold fundamental group of a cusp-decomposable orbifold is actually virtually isomorphic to the fundamental group of a, maybe different, cusp-decomposable orbifold.

\end{teo}

These results are inspired by those proven for certain higher dimensional graph manifolds studied by Frigerio, Lafont and Sisto.

An example of decomposition coming from an operation on the manifold is the canonical geometric decomposition for closed non-positively curved manifolds along flat codimension $1$ submanifolds, described by Leeb and Scott. It somewhat generalizes the JSJ decomposition in the $3$-dimensional case.

Fundamental groups of cusp-decomposable manifolds turn out to be \emph{relatively hyperbolic}, which greatly simplifies arguments in this case. We use here the definition of relative hyperbolicity using asymptotic cones following Dru\c{t}u and Sapir. Our exposition relies on some insight on the Model Theory underlying the asymtptotic cone construction. This exposition choice is uncommon in the literature on the subject.

Relative hyperbolicity no longer holds for non-positively curved manifolds. We explore here the large scale geometric structure of non-positively curved manifolds with non-trivial Leeb-Scott decomposition and state what is missing to prove a quasi-isometric rigidity result in higher dimensions, similar to the one of Kapovich and Leeb for dimension $3$. More precisely, we propose some conjectures on the quasi-isometric rigidity of the Leeb-Scott decomposition, and we prove some results that could prove useful to establish our conjectures.

In Section 1 we review the theory necessary to this work. In particular, in Subsection 1.3 we describe the model-theoretic tools necessary to introduce non-standard universes, which we will use to construct asymptotic cones in the following subsection.

In Section 2 we will prove some technical results on negatively curved spaces which will turn useful in the proof of connectedness results in later sections.

In Section 3 we present the results on quasi-isometry for universal covers of cusp-decomposable manifolds. The exposition goes through four subsections. The first one is introductory and describes the construction of cusp-decomposable manifolds. The second and the third ones are more geometric in nature and prove the main technical results which will be used in the fourth subsection in the proof of the main theorems.

Section 4 is an exposition on what is known and what is still unknown on non-positively curved manifolds. After a short introduction, their large scale structure is the object of the second subsection, while the third one is dedicated to some conjectures, whose plausibility derives from similar results in analogous settings.

The most studied morphisms in this settings are quasi-isometric embeddings, i.e. maps $f$ between metric spaces $(E_1,d_1)$ and $(E_2,d_2)$ for which there are constants $K \geq 1$ and $c \geq 0$ such that the inequality

\[ K^{-1}d_1(p,q) - c \leq d_2 (f (p), f (q)) \leq Kd_1(p,q) + c \]

holds for every $p$, $q$ in $E$.

A quasi-isometric embedding is a quasi-isometry if there exists a quasi-isometric embedding $g \colon E_2 \to E_1$ such that the compositions $f \circ g$ and $g \circ f$ are at a bounded distance from the identity maps of $E_2$ and $E_1$ respectively.

If a group $G$ is generated by a finite set $S$, one can define $d (x,y)$ for $x$, $y$ in $G$ to be the minimal length of a string of elements of $S$ and inverses of elements of $S$ that represents $x^{-1}y$. This turns out to be a distance in $G$, invariant by left multiplication, and well defined up to bi-Lipschitz equivalence, hence quasi-isometry, if one changes the generating set. This is the starting point of Geometric Group Theory. One of the questions it tries to answer is whether certain algebraic properties of groups, or geometric properties of spaces on which these groups act nicely, are preserved under quasi-isometry.

A group is quasi-isometric to a finite index subgroup, and to any of its extensions by a finite group. These two steps generate the relation of virtual isomorphism between groups, and it follows that virtually isomorphic groups cannot be distinguished by quasi-isometries.

We can then define a class of groups to be \emph{quasi-isometrically rigid} if any group quasi-isometric to a group of the class is virtually isomorphic to a possibly different group in the class. We also say that a group is \emph{quasi-isometrically rigid} if any group quasi-isometric to it is virtually isomorphic to it.

In this thesis we address in particular the question of quasi-isometric rigidity for families of fundamental groups of certain manifolds having a canonical geometric decomposition. This decomposition comes either from the construction of the manifold, or from some operation performed on the manifold.

An example of the first case are cusp-decomposable manifolds defined by Nguy\^en-Phan. They are constructed by gluing along affine homeomorphisms of the boundary manifolds obtained by cutting away cusps of complete finite volume negatively curved locally symmetric manifolds. The precise definition will be given in Section 3. For this class of manifolds I present original results, the main of which are the following:

\begin{teo}

Let $\Gamma$ be a group quasi-isometric to the fundamental group of a cusp-decomposable manifold $M$. Then $\Gamma$ is virtually isomorphic to the fundamental group of a cusp-decomposable orbifold.

\end{teo}

\begin{teo}

The class of orbifold fundamental groups of cusp-decomposable orbifolds is quasi-isometrically rigid, i.e. every group quasi-isometric to an orbifold fundamental group of a cusp-decomposable orbifold is actually virtually isomorphic to the fundamental group of a, maybe different, cusp-decomposable orbifold.

\end{teo}

These results are inspired by those proven for certain higher dimensional graph manifolds studied by Frigerio, Lafont and Sisto.

An example of decomposition coming from an operation on the manifold is the canonical geometric decomposition for closed non-positively curved manifolds along flat codimension $1$ submanifolds, described by Leeb and Scott. It somewhat generalizes the JSJ decomposition in the $3$-dimensional case.

Fundamental groups of cusp-decomposable manifolds turn out to be \emph{relatively hyperbolic}, which greatly simplifies arguments in this case. We use here the definition of relative hyperbolicity using asymptotic cones following Dru\c{t}u and Sapir. Our exposition relies on some insight on the Model Theory underlying the asymtptotic cone construction. This exposition choice is uncommon in the literature on the subject.

Relative hyperbolicity no longer holds for non-positively curved manifolds. We explore here the large scale geometric structure of non-positively curved manifolds with non-trivial Leeb-Scott decomposition and state what is missing to prove a quasi-isometric rigidity result in higher dimensions, similar to the one of Kapovich and Leeb for dimension $3$. More precisely, we propose some conjectures on the quasi-isometric rigidity of the Leeb-Scott decomposition, and we prove some results that could prove useful to establish our conjectures.

In Section 1 we review the theory necessary to this work. In particular, in Subsection 1.3 we describe the model-theoretic tools necessary to introduce non-standard universes, which we will use to construct asymptotic cones in the following subsection.

In Section 2 we will prove some technical results on negatively curved spaces which will turn useful in the proof of connectedness results in later sections.

In Section 3 we present the results on quasi-isometry for universal covers of cusp-decomposable manifolds. The exposition goes through four subsections. The first one is introductory and describes the construction of cusp-decomposable manifolds. The second and the third ones are more geometric in nature and prove the main technical results which will be used in the fourth subsection in the proof of the main theorems.

Section 4 is an exposition on what is known and what is still unknown on non-positively curved manifolds. After a short introduction, their large scale structure is the object of the second subsection, while the third one is dedicated to some conjectures, whose plausibility derives from similar results in analogous settings.

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