Résumés 3manifolds

The rank of a group is the minimal number of elements needed to generate it. We’ll briefly discuss rank in general before turning to 3-manifold groups. There, topics may include:

1. the relationship between rank and Heegaard genus
2. work of Abert-Nikolov relating the growth of rank in a tower of covers to the « cost » of a certain measured equivalence relation,
3. my work with Souto on ranks of mapping tori with large curve complex translation distance, and on geometric decompositions of thick hyperbolic 3-manifolds with bounded rank.

A virtual property of a 3-manifold group is a property satisfied by a finite index subgroup. It can alternatively be thought of as a property satisfied by the finite-sheeted cover of the manifold. The study of such properties has led to important developments in 3-manifold topology and geometry. In this mini-course, we will introduce separability and related virtual properties with a focus on the arithmetic perspective.

After sketching some basic 3-manifold topology, I will discuss Thurston’s geometric viewpoint on 3-manifolds, including the statement of Perelman’s Geometrization Theorem. These results lead to the ubiquitous role of hyperbolic geometry in answering purely topological questions in dimension 3. I will then discuss some algorithmic and group-theoretic consequences of Geometrization, including solvability of the homeomorphism problem and residual finiteness of fundamental groups of compact 3-manifolds. I will also demonstrate now hyperbolic structures on 3-manifolds can be found in practice and used for large-scale computational experiments.

The focus of this minicourse will be certain conditions that conjecturally imply that a group is the fundamental group of a compact 3-manifold. I will discuss the Cannon Conjecture (that a word-hyperbolic group whose Gromov boundary is a 2-sphere has a finite-index subgroup which is a 3-manifold group), a relative version of the Cannon Conjecture, and a conjecture of Wall which states that a finitely generated PD(3) group is a 3-manifold group. Connections between these questions will be discussed, as well as criteria which imply them in certain cases.

A basic problem in low dimensional topology is to understand the 2-complexes with a given fundamental group G. I will explain how this problem can be studied using a division algorithm in the group ring of G and describe a situation in which such an algorithm is available. Additional applications include lower bounds on complexity of cell-decompositions of hyperbolic manifolds.

I will address the (somewhat mysterious) connection between arithmeticity and geometry, and in particular the question: can you read the arithmeticity of a manifold from its geometry? The talk will be based on a joint work with Fisher, Miller and Stover, but my main aim will be to survey the subject, rather than give fully detailed proofs.

Takahasi and Higman independently proved that any ascending chain of subgroups of constant rank in a free group must stabilize. Kapovich and Myasnikov gave a proof of this fact in the language of graphs and Stallings folds. Motivated by Kapovich and Myasnikov’s proof N. Lazarovich and I provide two new proofs of the constant-rank ascending chain condition (cracc) for closed surface groups. Extending the techniques of these two new proofs, we establish cracc for free subgroups of constant rank in a closed (or finite-volume hyperbolic) 3-manifold group. Hyperbolic geometry, geometrization, and JSJ decompositions all play a role in the proofs.

In a recent joint paper with Misha Belolipetsky, Sasha Kolpakov and Leone Slavich we developed a large industry connecting geometry and arithmeticity of hyperbolic orbifolds and manifolds. We introduce a new class of the so-called fc-subspaces (which correspond to finite symmetries of finite covers, or are fixed point sets of finite subgroups of the commensurator) and use them to formulate an arithmeticity criterion for hyperbolic orbifolds: a hyperbolic orbifold is arithmetic if and only if it has infinitely many fc-subspaces.  We also show that immersed totally geodesic m-dimensional suborbifolds of n-dimensional arithmetic hyperbolic orbifolds are fc-subspaces whenever m \ge \floor{n/2}, and we provide examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. One of the key results of our paper is an algebraic characterization of immersions of arithmetic hyperbolic orbifolds into one another. Thus, this machinery of fc-subspaces also provides an efficient approach and solution to the problem of detecting arithmeticity of a hyperbolic n-manifold only through its geometry (this problem has remained open in general for several decades). Time permitting, I will also discuss a few applications of fc-subspaces to reflection groups and hyperbolic link complements; this is based on the upcoming joint preprint with Dmitry Guschin and Andrei Vesnin.

Certain 3-manifold groups are uniquely determined among all finitely generated, residually finite groups by the set of their finite images. In this talk, I shall explain why this is so, and then describe infinitely many such groups. I shall also describe groups that are profinitely rigid among finitely presented groups, but not among finitely generated groups.

The finite quotients of the fundamental group of a 3-manifold are the deck groups of its regular covers. These finite quotients are organized to obtain the profinite completion of a 3-manifold group. We discuss how to leverage properties of mapping class groups of finite-type surfaces in order to study profinite completions of the fundamental groups of fibered hyperbolic 3-manifolds.

We describe a straightforward strategy for producing dense embeddings of certain amalgamated products into simple linear Lie groups. The geometric nature of the strategy allows one to dictate properties of the output dense embedding; for instance, it is sometimes possible to ensure a lack (or presence) of nontrivial unipotents in the image, or arrange for each matrix in the image to have algebraic entries. We use this approach to construct dense embeddings of some hyperbolic manifold groups. This is joint work with Konstantinos Tsouvalas.

Let K and J be two knots such that there exists an epimorphism of fundamental group of S^3\K onto the fundamental group of S^3\ J, then the genus of J is at least the genus of K.
We discuss a generalization of this problem for general 3-manifolds and we prove special cases of this generalization. We state several conjectures on fundamental groups of knots, links and 3-manifolds, which would imply a complete solution to Simon’s conjecture.

I will discuss how one can recogise virtually free-by-cyclic groups among hyperbolic compact special groups solely by using homology. (Joint with Marco Linton.)

In this talk, I will discuss almost profinite rigidity of finite-volume hyperbolic 3-manifold groups and its further development. Volume, invariant trace fields, arithmeticity are some most important geometric quantities/properties, and I’ll discuss the question about their profinite invariance.

Thurston’s Hyperbolic Dehn filling theorem in 3-manifolds generalizes to the setting of relatively hyperbolic groups.  In this setting the analog of a geometrically finite subgroup is a relatively quasiconvex subgroup.  We give conditions under which quasiconvexity of a subgroup persists under a Dehn filling, as well as related « weak separability » results.  Despite the somewhat abstract setting (relatively hyperbolic groups) there are applications to 3-manifold topology. This is joint work with Daniel Groves.

Associated to any finite volume hyperbolic 3-manifold M are certain topological invariants called the invariant trace field and invariant quaternion algebra. When M has cusps, of interest is the behavior of these invariants under Dehn filling of the corresponding manifold. In the setting where M is a hyperbolic knot complement in the 3-sphere, work of Chinburg, Reid, and Stover shows that if the Alexander polynomial satisfies a particular number theoretic property, then there are significant restrictions on the behavior of these invariants under Dehn filling. This relationship functions through a certain algebro-geometric object living over the canonical component called an Azumaya algebra. The goal of this talk will be to survey this connection as well as discuss recent results extending this to the case of once-punctured torus bundles. I will aim to be as friendly to the audience and expository as possible, especially to those who do not particularly care for number theory and/or algebraic geometry.

The Bowditch boundary of a group G relatively hyperbolic with respect to a family P of subgroups is a compactum on which G acts and that is an invariant of the pair (G,P). Groups of hyperbolic knots with respect to their peripheral subgroups provide the most basic examples of relatively hyperbolic group pairs.

A natural and very general question in this context is the following: given a compactum K and assuming that it is a Bowditch boundary of some pair (G,P), what can we deduce about G and P from K?

I will present a joint work with Peter Haïssinsky and Genevieve Walsh, in which we address this question under the hypothesis that K is planar, that is embeds into the 2-sphere, and the action of G estends to the whole sphere. We are specifically interested in the case where K is a Schottky set. We show that Schottky sets come in three combinatorial flavours for two of which it is possible to give a description of the structure of G.

The Geometrization Theorem states that every closed 3-manifold can be canonically decomposed into pieces that each have one of eight Thurston geometries. In joint preliminary work with D. Cooper and L. Mavrakis, we show that each of the eight geometries is “locally combinatorially defined” (LCD), which we use to describe a new algorithm for finding a geometric decomposition of a 3-manifold and a new way to build a random 3-manifold. In this talk, I will provide examples to motivate the definition of a family of spaces being LCD. I will then explain how branched 3-manifolds play a key role in our approach towards this problem by sketching the proof of our results for a few geometries.

There is a natural interest in hyperbolic manifolds of low volume, and this talk addresses dimension four. As opposite to dimension n = 3 where Thurston’s hyperbolic Dehn filling holds, for n > 3 the volume spectrum is discrete, and there is at most a finite number of hyperbolic n-manifolds with bounded volume (Wang’s finiteness).

Computing the number of hyperbolic 4-manifolds of given small (even minimal) volume appears nowadays far from reach. Counting such manifolds up to commensurability seems less unrealistic, at least by restricting the count to arithmetic manifolds.

We will give an overview of the known examples of low-volume hyperbolic 4-manifolds, with particular attention to the construction of some cusped manifolds by means of a remarkable family of polytopes discovered in 2010 by Kerckhoff and Storm. This will include some results obtained in joint works with Martelli and Slavich.

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings.

We generalize the structure and the construction of the JSJ decomposition, to study the automorphisms of a (colorable) hierarchically hyperbolic group that satisfies some weak acylindricity conditions. The object that we construct can be viewed as a higher rank JSJ decomposition. Like the JSJ decomposition of a hyperbolic group it encodes information on the algebraic structure of the automorphism group and on the dynamics of individual automorphisms.

There has recently been much work focused on the problem of distinguishing the fundamental groups of 3-manifolds by using their finite quotients. Many properties of 3-manifolds have been shown to be detectable in this way, and certain 3-manifolds are known to be completely determined by their finite quotients. A few 3-manifold groups are known to be profinitely rigid, they are determined among all finitely generated, residually finite groups by their collection of finite quotients. I will discuss some recent work which describes how the finite quotients of a group can detect its linear representations and explain what work remains to be done to show more 3-manifold groups are profinitely rigid.

We give a complete characterization on the separability of a subgroup of a 3-manifold group, in terms of the graph-of-group structure induced by the torus decomposition. We will also talk about two applications of this characterization on other aspects of 3-manifold groups: subgroup distortion and Grothendieck rigidity of 3-manifold groups. The subgroup distortion part is joint with Hoang Thanh Nguyen.

For a 3-manifold M, we consider the problem of lifting subgroups G < Mod(M) of the mapping class group Mod(M)=Diff(M)/isotopy under the natural surjection Diff(M) → Mod(M). We will discuss some different aspects of this problem and give an answer for subgroups generated by sphere twists. This is joint work with Lei Chen.

Let p be a prime number and d a positive integer. Cuoco and Monsky generalized Iwasawa’s class number formula for Zp^d-extensions of number fields, describing the growth of p-torsions. In the spirit of arithmetic topology, we establish similar results for Zp^d-covers of links consisting of rational homology 3-spheres. More precisely, we compare several Iwasawa modules and refine our previous results to a system of (Z/p^nZ)^d-covers which is not necessarily derived from a Z^d-cover. We also give a remark on p-adic torsions.

For a hyperbolic group, we define « drilling » the group along a maximal cyclic subgroup.  This is an analog of drilling a hyperbolic manifold along a geodesic.  We show that, under suitable hypotheses, drilling a hyperbolic group with boundary S^2 results in a relatively hyperbolic group with relative boundary S^2  We also give an application to the Cannon Conjecture.  This is joint work with  D. Groves, P. Haissinsky, J. Manning, D. Osjada and A. Sisto.