Seminars (NPCW01)

A cornerstone of the study of mapping class groups is the Nielsen--Thurston classification theorem. I will outline a polynomial-time algorithm that determines the Nielsen--Thurston type and the canonical curve system of a mapping class. Our approach uses the action on the curve complex. Time permitting, I shall discuss the conjugacy problem for the mapping class group. This is joint work with Mark Bell.

Co-authors: Rémi Coulon (Université de Rennes 1), Arnaud Hilion (Aix-Marseille Université), Gilbert Levitt (Université de Caen)
 
Given a torsion-free hyperbolic group G and an outer automorphism \Phi of G, we investigate the possible growth types of conjugacy classes of G under iteration of \Phi.

We show that groups acting freely on CAT(0) square complexes satisfy uniform exponential growth. This is joint work with Aditi Kar.

We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers. Joint work with Benjamin Beeker.

I will discuss a few results about the bounded cohomology of an acylindrically hyperbolic group, focusing on quasimorphisms for concreteness. More specifically, the results relate the bounded cohomology of an acylindrically hyperbolic group to that of its hyperbolically embedded subgroups, which are special subgroups that generalise peripheral subgroups of relatively hyperbolic groups.

Co-authors: Giuliano Basso (ETH Zurich), Dominic Descombes (ETH Zurich), Benjamin Miesch (ETH Zurich)
 
A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. The existence of a geodesic bicombing satisfying a suitable convexity condition may be viewed as a weak (but non-coarse) global notion of non-positive curvature that allows for non-unique geodesics. The talk will give a survey of recent results on group actions on spaces with such a structure.

We give a fresh view on Moebius geometry and show that the ideal boundary of a negatively curved space has a natural Moebius structure. We discuss various cases of the interaction between the geometry of the space and the Moebius geometry of its boundary. We discuss an approach how the concept of Moebius geometry can be generalized in order that it is usefull for the boundaries of nonpositively curved spaces like higher rank symmetric spaces, products of rank one spaces or cube complexes. In particular we describe a Moebius geometry on the Furstenberg boundary of a symmetric space.

Admitting a "nice" action on a hyperbolic space has strong algebraic and geometric consequences for a group. For this talk we will focus on replacing "nice" by "non-trivial acylindrical". The goal of the talk is to open and discuss a number of interesting and important recognition problems for non-trivial acylindrical actions of groups on hyperbolic spaces and to discuss the relation between these problems and the existence of coarse negative curvature in the group. This is a summary of current progress on many projects with a variety of co-authors.

Co-authors: Pallavi Dani (Louisiana State University), Anne Thomas (University of Sydney)
 
I will describe the abstract commensurability classification within a class of hyperbolic right-angled Coxeter groups. I will explain the relationship between these groups and a related class of geometric amalgams of surface groups, and I will highlight the differences between the quasi-isometry classification and abstract commensurability classification in this setting. This is joint work with Pallavi Dani and Anne Thomas. 

The outer automorphism groups of right-angled Artin groups (RAAGs) give a way to build a bridge between GL(n,Z) and Out(Fn). We will investigate certain properties of these groups which could be described as "vastness" properties, and ask if it possible to build a boundary between those which are "vast" and those which are not. One such property is as follows: given a group G, we say G has all finite groups involved if for each finite group H there is a finite index subgroup of G which admits a map onto H. From the subgroup congruence property, it is known that the groups GL(n,Z) do not have every finite group involved for n>2. Meanwhile, the representations of Out(Fn) given by Grunewald and Lubotzky imply that these groups do have all finite groups involved. We will describe conditions on the defining graph of a RAAG that are necessary and sufficient to determine when it's outer automorphism group has this property. The same criterion also holds for other properties, such as SQ-universality, or having a finite index subgroup with infinite dimensional second bounded cohomology. This is joint work with V. Guirardel.

(Joint with Eric Swenson)
It is known that if the boundary of a 1-ended hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0) groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G
then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated
by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.

 The question of whether a finitely generated or presented group is linear has been looked at in various contexts, as has the existence of well behaved geometric actions of such groups by isometries on metric spaces although there is no single universally accepted definition of what such a well behaved action should be.
 
However at first sight there appears little connection between these two concepts, for instance one might consider some infinite simple groups or Baumslag-Solitar groups to make this point. In this talk we will first look at such groups to illustrate known results, before giving examples and evidence for the implication "linear implies good behaviour geometrically" if we use the appropriate notion of linearity.