Geometric Group Therapy Seminar

Wise asked whether all non-positively curved groups satisfy the following alternative: Given two elements of the group, can we always find powers that either commute or generate a free group? In this talk, I will present what is known about this alternative, as well as some related questions.

Analytic torsion is an invariant of Riemannian manifolds introduced by Ray and Singer in the 70's, defined as an analytic analogue of the topological invariant Reidemeister torsion. More recently, Bergeron and Venkatesh used analytic torsion and its analogy to Reidemeister torsion to describe the growth of torsion in the homology of cocompact arithmetic groups. This has number-theoretic significance by the Ash conjecture, partially proven by Scholze, relating torsion in the homology of arithmetic groups to the existence of Galois representations. One of the main ingredients is an approximation theorem, expressing L^2-torsion as a limit using analytic torsion. Better control of this approximation theorem should yield improved bounds on the homology of arithmetic groups. 

In this talk, we present an approximation theorem for arithmetic groups that are not cocompact. We furthermore provide a rate of convergence result in this setting, which conjecturally implies stronger bounds on torsion homology.

We introduce the notions of asymptotic rank and injective hulls before investigating a coarse version of Dress’ 2(n+1)-inequality characterising metric spaces of combinatorial dimension at most n. This condition, referred to as (n,δ)-hyperbolicity, reduces to Gromov's quadruple definition of δ-hyperbolicity for n=1. The ℓ∞ product of n δ-hyperbolic spaces is (n,δ)-hyperbolic and, without further assumptions, any (n,δ)-hyperbolic space admits a slim (n+1)-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. Using tools from recent developments in geometric group theory, we look at some examples and show that every Helly group and every hierarchically hyperbolic group of asymptotic rank n acts geometrically on some (n,δ)-hyperbolic space. Finally,  we relate (2,0)-hyperbolicity to the Nagata dimension, a variation of Gromov's asymptotic dimension with important applications related to extendibility of Lipschitz maps, via locally finite Helly graphs of combinatorial dimension at most two. Joint work with Urs Lang.

Starting with questions about units and idempotents in group rings the talk will motivate the statement of the Farrell—Jones Conjecture in algebraic K-theory. Time permitting I will try to outline how geodesic flow is used in proofs of cases of this conjecture.

Graphical small cancellation is a powerful extension of classical small cancellation theory, used to construct groups with exotic properties. It is expected to have close connections with various notions of nonpositive curvature, as in the classical setting. In this talk, we will discuss the CAT(0) property of graphical C(3)-T(6) small cancellation complexes. By a result of Osajda and Przytycki, this immediately implies the Tits alternative for groups acting freely on such complexes.

In 2016 Świątkowski examined non-elementary graphs of groups with finite edge groups and all vertex groups being CAT(0). He showed that the fundamental group of said graphs of groups admits a CAT(0)-boundary which is homeomorphic to a dense amalgam of CAT(0)-boundaries of vertex groups. In this talk we will also investigate the behavior of boundaries of such a fundamental groups and observe a result that is almost the converse of the aforementioned theorem. Namely, we will see that any CAT(0)-boundary of the discussed graph of groups is homeomorphic to a dense amalgam of limit sets of vertex groups. This result is an offshoot of the speaker’s bachelor thesis written under supervision of Jacek Świątkowski.

In the last decades, locally indicable groups have emerged in many works related to known problems in topology, algebra, and geometry. These groups are characterized by the property that every non-trivial finitely generated subgroup admits an epimorphism onto the integers. In particular, they appear in problems concerning asphericity, orderability, and equations over groups.

In the early 2000s, Wise introduced a topological variant of non-positive curvature for 2-complexes: the non-positive immersion property. This notion is closely related to local indicability and is also used to study the coherence and cohomological properties of the fundamental groups of 2-complexes. Recently, this concept has regained relevance due to its deep connection with Bausmlag's conjecture, which was recently proven by A. Jaikin-Zapirain and M. Linton.

In this talk, I will present new methods, developed in joint work with Gabriel Minian, to study the local indicability of groups that admit generalized Wirtinger presentations. We then apply our methods to study the non-positive immersion property for the 2-complexes associated to these presentations. This provides a large family of concrete examples of 2-complexes (which can be described algorithmically) with the non-positive immersion property.

In collaboration with David Gao and Greg Patchell, we developed a theory of soficity for actions of groups on sets and more generally graphs. I will describe various facets of this theory with applications. 

Bi-exactness is an analytic property of groups introduced by Ozawa. This property provided a fundamentally new way to prove primeness of group von Neumann algebras and has been intensively studied from operator algebraic perspective. On the other hand, not many examples of bi-exact groups were found so far. I'll talk about my endeavor to investigate bi-exactness from group theoretic perspective.

CAT(0) cube complexes form a class of cell complexes with some very nice combinatorial structures, and they have proved to be a useful tool for studying nonpositively curved groups. In this talk, I will discuss some interactions between CAT(0) cube complexes and the large-scale geometry of groups.

I will discuss the relationship between the class of biautomatic groups and various notions of non-positive curvature.  I intend to introduce various examples of biautomatic and non-biautomatic groups, to discuss results ranging from the introduction of biautomatic groups in late 1980s to a paper that appeared on the arXiv on 9 December 2024, and to provide more questions than answers.

In this talk, we will show that the 2-complex associated to the presentation ⟨a,b∣b,bab⁻¹a⁻²⟩ has contracting nonpositive immersions and positive maximal irreducible curvature. This example shows that the contracting non-positive immersions property is not equivalent to the notion of nonpositive irreducible curvature, answering a question raised by H. Wilton.

Small cancellation theory was originally developed to help address the decision problem in group theory. Its assumptions are intuitively related to nonpositive curvature but do not directly imply it in a metric sense. In this talk, we’ll introduce small cancellation conditions for 2-complexes and explore their connections to group theory. Our main goal is to present Duda’s result that every C(3)-T(6) small cancellation complex admits a CAT(0) metric.

During this seminar we will continue the topic from last week. We will discuss two counterexamples for conjectures stated by Levitt and McCammond. Lastly we will draw some conclusions and say a little bit about future directions of research.

In this talk we will talk over some of the results from R. Levitt and J. McCammond paper "Triangles, Squares and Geodesics". Firstly, we will discuss biautomaticity of CAT(0) cubical groups and systolic groups, which became the motivation for the paper mentioned above. Then we are going to define Gersten-Short geodesics and analyse the attempt of proving biautomaticity of CAT(0) triangle-square groups by Levitt and McCammond. Lastly we are going to see two counterexamples which disallow such geodesics from satisfying the two-sided fellow traveler property.

Automatic and biautomatic groups were introduced in the 80s and described in greater details by David B. A. Epstein et al. in the book "Word Processing in Groups". In the talk we will see the definition as well as examples and non-examples of (bi)automatic groups. We will also discuss some properties and theorems of the aforementioned groups.

This is a brief introduction to the mapping class group of a surface.  One of the main goals is to give a description of the Nielsen-Thurston classification of mapping classes.

These talks are a preface to Jing Tao's talk at the Groups and Operator Algebras Seminar.

The Charney-Davis strict hyperbolization is a procedure that turns polyhedra into spaces of negative curvature, while preserving some topological features. It has been used to construct examples of aspherical manifolds that exhibit unexpected features, despite having negative curvature. One may expect the fundamental groups of these manifolds to display strange features as well. On the other hand, we show that these groups admit nice actions on CAT(0) cube complexes. As an application, we obtain new examples of negatively curved Riemannian manifolds whose fundamental groups are linear over the integers, virtually special, and algebraically fibered.