Abstracts

Giles Gardam (Münster): One-relator groups and the Haagerup property

It is open whether every Group admitting a presentation with a single relator has the Haagerup property (a.k.a. Gromov's a-T-menability). I will outline the theory of one-relator Groups relevant to the Haagerup property, including both why establishing this property is easy in certain cases and some joint work with Dawid Kielak demonstrating the difficulty of the problem in general. We conclude with some cause for optimism.

Nima Hoda (ENS Paris): Crystallographic Helly groupsg

A Helly graph is a graph in which the metric balls form a Helly
family: any pairwise intersecting collection of balls has nonempty
total intersection.  A Helly group is a group that acts properly and
cocompactly on a Helly graph.  Helly groups simultaneously generalize
hyperbolic, cocompactly cubulated and C(4)-T(4) graphical small
cancellation groups while maintaining nice properties, such as
biautomaticity.  I will show that if a crystallographic group is Helly
then its point group preserves an L^{\infinity} metric on \R^n.  Thus
we will obtain some new nonexamples of Helly groups, including the
3-3-3 Coxeter group, which is a systolic group.  This answers a
question posed by Chepoi during the recent Simons Semester on
Geometric and Analytic Group Theory in Warsaw.

Markus Lohrey (Siegen):  The compressed word problem for groups

The word problem is arguably the most important computational problem in

combinatorial group theory. For a finitely generated group G the word problem
asks whether a given word over the group generators evaluates to the group
identity. In recent years, it turned out to be useful to consider also variants of the word
problem, where the input word is given in a kind of compressed form. In particular,
so called straight-line programs have attained a lot of attention. A straight-line program
is a context-free grammar that evaluates to a unique word. The latter can be much
longer than the number of symbols in the grammar. In this sense, a straight-line program
can be viewed as a compressed representation of a string. This leads to the compressed
word problem for a finitely generated group G. The input consists of a straight-line program
that produces a word w over the group generators, and the question is whether w evaluates
to the group identity of G. The compressed word problem is also useful, if one is mainly
interested in the (classical) word problem. For instance, the word problem for the automorphism
group of a group G can be easily reduced to the compressed word problem for G. This was
the main idea for the first (and to the knowledge of the speaker only) polynomial time algorithm
for the automorphism group of a free group. In my talk I will give an overview on existing
results for compressed word problems. We will see that for many groups the compressed word
problem can be solved in polynomial time (e.g. finitely generated nilpotent groups, virtually
special groups, hyperbolic groups). On the other hand, there exist groups, where the compressed
word problem is much harder to solve than the classical word problem. Examples for this are the

Grigorchuk group and Thompson’s group F.

Annette Karrer (KIT): Morse boundaries of right-angled Coxeter groups

Morse boundaries are quasi-isometry invariants of proper metric spaces which capture how similar a space is to a hyperbolic space. In this talk, we explain how to construct right-angled Coxeter Groups (RACGs) with totally disconnected Morse boundaries. Moreover, we will study interesting situations in which large connected components occur. Important tools are Splitting over sub-RACGs, itineraries in the corresponding Bass-Serre trees and characterisations of rank-one isometries in RACGs.

Emily Stark (University of Utah): Action rigidity for free products of hyperbolic manifold groups

The relationship between the large-scale geometry of a
group and its algebraic structure can be studied via three notions: a
group's quasi-isometry class, a group's abstract commensurability
class, and geometric actions on proper geodesic metric spaces. A
common model geometry for groups G and G' is a proper geodesic metric
space on which G and G' act geometrically. A group G is action rigid
if every group G' that has a common model geometry with G is
abstractly commensurable to G. For example, a closed hyperbolic
n-manifold is not action rigid for all n at least three. In contrast,
we show that free products of closed hyperbolic manifold groups are
action rigid. Consequently, we obtain the first examples of Gromov
hyperbolic groups that are quasi-isometric but do not virtually have
a common model geometry. This is joint work with Daniel Woodhouse.

Daniel Woodhouse (University of Oxford): Quasi-isometric rigidity of some graphs of groups

We present (almost complete) work in progress, joint with Sam
Shepherd, showing that certain graphs of free groups with cyclic edge
groups are quasi-isometrically rigid.