## MINI-COURSES

### Ian BIRINGER (Boston College)

#### Rank of 3-manifold groups

## Abstract

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:

- the relationship between rank and Heegaard genus
- work of Abert-Nikolov relating the growth of rank in a tower of covers to the “cost” of a certain measured equivalence relation,
- 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.

### Michelle CHU (University of Minnesota)

#### Virtual properties of 3-manifold groups

## Abstract

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.

### Nathan DUNFIELD (University of Illinois Urbana-Champaign)

#### Geometrization and its consequences

## Abstract

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.

### Daniel GROVES (University of Illinois at Chicago)

#### 3-manifold groups?

## Abstract

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.

## TALKS

### Grigori AVRAMIDI (Max Planck Institute, Bonn)

#### Division, group rings, and negative curvature

## Abstract

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.

### Uri BADER (Weizmann Institute of Science)

#### The geometry of arithmetic manifolds

## Abstract

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.

### Edgar BERING (San José State University)

#### Ascending chains of free groups in 3-manifold groups

## Abstract

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.

### Nikolay BOGACHEV (University of Toronto)

#### Arithmeticity and fc-subspaces of hyperbolic manifolds

## Abstract

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.

### Martin BRIDSON (University of Oxford)

#### Profinite rigidity, triangle groups, and Seifert fibred spaces

## Abstract

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.