Conference "Topological and Homological Methods in Group Theory 2018"

• Faculty of Mathematics
• Bielefeld University

Programme

All talks take place in lecture hall H6.

Titles and abstracts

Mladen Bestvina

Say a group G is EPQT (for Equivariant embedding in Product of Quasi Trees) if G acts by isometries on a finite product of quasi-trees (preserving the product structure) so that orbit maps are QI embeddings. This notion is a strong form of finiteness of asymptotic dimension.

Thm 1: Every residually finite hyperbolic group is EPQT.

Thm 2: (Hamenstädt): Mapping class groups are EPQT.

The proofs use the construction of projection complexes. This work is joint with Ken Bromberg and Koji Fujiwara.

Martin Bridson

Developments in geometry and low dimensional topology have given renewed vigour to the following classical question: to what extent do the finite images of a finitely presented group determine the group? I'll survey what we know about this question in the context of $3$-manifolds, and I shall present recent joint work with McReynolds, Reid and Spitler showing that the fundamental groups of certain hyperbolic orbifolds are distinguished from all other finitely generated groups by their finite quotients.

Pierre-Emmanuel Caprace

The group $\mathrm{Aut}(F_n)$ is a prominent character in geometric group theory. The goal of this talk is to advertise a larger group, namely the group of abstract commensurators of $F_n$, denoted by $\mathrm{Comm}(F_n)$. Bartholdi and Bogopolski have shown that $\mathrm{Comm}(F_n)$ is infinitely generated. A. Lubotzky has asked whether $\mathrm{Comm}(F_n)$ is simple. I will explain that $\mathrm{Comm}(F_n)$ is almost simple. The relative commensurator of F_n in the automorphim group of its Cayley tree, and commensurator groups of surface groups, will also be mentioned, and the relevance of non-discrete topological groups will be emphasized.

Elia Fioravanti

Median spaces are a generalisation of CAT(0) cube complexes that allows non-discreteness; one-dimensional examples are provided by real trees. We will discuss an extension to median spaces of a known superrigidity result of Chatterji-Fernos-Iozzi for actions on CAT(0) cube complexes.

The generalisation is by no means trivial, as certain pathologies may arise and many combinatorial tools are not available in this context. It provides surprising fixed point properties for irreducible lattices in products of rank one simple Lie groups. These lattices cannot act nontrivially on a finite rank median space, even though their coarse geometry has strong median features and they do act properly and cocompactly on median spaces of infinite rank.

As a corollary, there are only finitely many conjugacy classes of homomorphisms from any such lattice to any (equivariantly) coarse median group. The latter form a large class of groups, which includes mapping class groups and most cubulated groups.

Based on: arxiv:1711.07737

Alessandra Iozzi

We start by recalling features of classical Teichmüller space of a topological surface. We explain the notion of "higher Teichmüller" by introducing the notion of maximal representations and of Hitchin component and we see how some of the classical features, such as the Thurston compactification, generalize to the "higher Teichmüller" setting. In the process we explain a structure theorem for geodesic currents. This is joint work with M. Burger, A. Parreau and B. Pozzetti.

Tadeusz Januszkiewicz

I will discuss several classes of cubical complexes arising outside Geometric Group Theory proper:

• real toric varieties,
• iterated blowups,
• generalised tridiagonal manifolds.

Each case brings its own specific questions, and each case has its "complex version".

Steffen Kionke

In the theory of $L^2$-invariants methods from functional analysis are used to generalize classical homological invariants of finite simplicial complexes to infinite simplicial complexes with a cocompact proper action of a group. Most prominently, the $L^2$-Betti numbers are defined by replacing the dimension theory of vector spaces by the dimension theory of the group von Neumann algebra.

In this talk we introduce and discuss new $L^2$-invariants: the $L^2$-multiplicities. These invariants can be defined for groups together with a finite group of automorphisms. We discuss an approximation theorem for $L^2$-multiplicities which generalizes Lück's approximation theorem for $L^2$ Betti numbers. In the end, we mention a character theoretic perspective on sofic groups which is related to $L^2$-multiplicites.

Jon McCammond

The n-strand braid group can be viewed as the fundamental group of the configuration space of n unordered points in a closed disk based at a configuration with all the points in the boundary of the disk. When viewed in this way, subgroups isomorphic to braid groups with fewer strands can be defined by specifying that some of the points remain fixed. This talk introduces an extension of this idea. A boundary braid is a braid that has a representative where a specified subset of the points remain in the boundary cycle of the disk but need not be fixed. Because the marked points do not necessarily return to their original positions, either pointwise or as a set, the collection of boundary braids with k marked strands merely forms a subgroupoid of the braid group rather than a subgroup. Our main result about these boundary braid subgroupoids is that they determine a geometrically interesting subcomplex of the dual braid complex, a conjecturally CAT(0) metric simplicial complex with a geometric braid group action. In particular, we prove a metric decomposition theorem for these boundary braid subcomplexes using a new type of orthoscheme configuration space of independent interest. This is joint work with Michael Dougherty and Stefan Witzel.

Beatrice Pozzetti

Bounded cohomology, introduced by Gromov in 1982 as a tool to study volumes of negatively curved manifolds, has been very powerful in tackling various rigidity questions. However, at least in the case of discrete groups, is still poorly understood and shows strange behaviors: I will discuss joint work with Frigerio and Sisto and with Frigerio, Franceschini and Sisto in which we show that the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional and not a Banach space.

Tomasz Prytuła

For an infinite discrete group $G$, the classifying space for proper actions $\underline{E} G$ is a proper $G$-CW-complex $X$, such that for every finite subgroup $F \subset G$ the fixed point set $X^F$ is contractible.

In a joint work with Nansen Petrosyan we describe a procedure of constructing new models for $\underline{E}G$ out of the standard ones, provided the action of $G$ on $\underline{E}G$ admits a strict fundamental domain. Our construction is of combinatorial nature, and it depends only on the structure of the fundamental domain. The resulting model is often much `smaller' than the old one, and thus it is well-suited for (co-)homological computations. Before outlining the construction, I shall give some background on the space $\underline{E}G$. I will also discuss some examples and applications in the context of Coxeter groups, graph products of finite groups and automorphism groups of buildings.

Alessandro Sisto

In 3-manifold theory, Dehn filling is an important construction of closed hyperbolic manifolds starting from finite-volume ones. There is a very useful algebraic version of such construction in the context of relatively hyperbolic groups, and this version has been used, for example, in the proof of the virtual Haken conjecture.

I will describe a method tha I developed with Groves and Manning to control the effect of Dehn filling on the boundary at infinity; the method involves describing the limit boundary as a limit, in an appropriate sense, of approximating spaces.

Considering the case of planar boundaries, this yields an application related to Cannon’s Conjecture. Namely, the (formally stronger) relative version of Cannon’s Conjecture can be reduced to the usual Cannon’s Conjecture.

I will also touch upon work in progress with Groves, Manning, Osajda, and Walsh about a "Dehn drilling" construction one can perform on hyperbolic groups with 2-sphere boundary. This can be thought of as the inverse operation of Dehn filling, and yields a relatively hyperbolic group with 2-sphere boundary.

Rachel Skipper

A group is of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We discuss some recent results about finiteness properties of Nekrashevych groups, a class of groups whose building blocks are self-similar groups and Higman-Thompson groups. Since these groups are often virtually simple and since finiteness properties are a quasi-isometry invariant, we use these results to build new examples of non-quasi-isometric simple groups.

This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.

Ignat Soroko

An interplay between algebra and topology goes in many ways. Given a space $X$, we can study its homology and homotopy groups. In the other direction, given a group $G$, we can form its Eilenberg-Maclane space $K(G,1)$. It is natural to wish that it is 'small' in some sense. If $K(G,1)$ space has $n$-skeleton with finitely many cells, then $G$ is said to have type $F_n$. Such groups act naturally on the cellular chain complex of the universal cover for $K(G,1)$, which has finitely generated free modules in all dimensions up to $n$. On the other hand, if the group ring $\mathbb{Z}G$ has a projective resolution $(P_i)$ of length $n$ where each module $P_i$ is finitely generated, then $G$ is said to have type $FP_n$. There have been many intriguing questions on whether classes $F_n$ and $FP_n$ are different, and some of them are still open. Bestvina and Brady gave first examples of groups of type $FP_2$ which are not finitely presentable (i.e. not of type $F_2$). In his recent paper, Ian Leary has produced uncountably many of such groups. Using Bowditch's concept of taut loops in Cayley graphs, we show that Ian Leary's groups actually form uncountably many classes up to quasi-isometry. This is a joint work with Ian Leary and Robert Kropholler.

Emily Stark

Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. If, in addition, the automorphism is fully irreducible, then work of Kapovich--Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to use the Cannon--Thurston map to construct embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. Along the way, I will illustrate how our methods distinguish free-by-cyclic groups which are the fundamental group of a 3-manifold. This is joint work with Yael Algom-Kfir and Arnaud Hilion.

Bin Sun

The notion of an acylindrically hyperbolic group was introduced by Osin as a generalization of non-elementary hyperbolic and relative hyperbolic groups. This class of groups not only contains plenty of interesting examples, for instance groups with deficiency at least two and outer automorphism groups of non-abelian free groups, but also has various nice algebraic, geometric and analytic properties so that useful tools such as Monod-Shalom rigidity theory, group theoretic Dehn surgery and small cancellation theory can be applied to yield beautiful results. By analyzing the induced action of an acylindrically hyperbolic group on the Gromov boundary of a hyperbolic space, we give a dynamical characterization of acylindrically hyperbolic groups. This result can be used to prove the acylindrical hyperbolicity of groups coming from dynamical actions. As an application, we prove that non-elementary convergence groups are acylindrically hyperbolic.

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