## 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.