Schedule

Pierre-Emmanuel Caprace (UCLouvain) -- 10:00-11:00

Title: Ascending chains of irreducible lattices in products of trees. 

Abstract: A classical result of Kazhdan-Margulis ensures that the set of covolumes of lattices in a semisimple Lie group is bounded below by a strictly positive constant. In particular every ascending chain of lattices is eventually constant. This contrasts with the case of lattices in Euclidean space, where the mesh of lattices can be arbitrarily small. The existence of ascending chains of lattices has also been observed in negatively curved spaces like trees or right-angled buildings. In this talk, I will first review known results, and then explain the construction of ascending chain of irreducible lattices in products of trees. Based on joint work with Justin Vast.

Nir Lazarovich (Technion) -- 11:30-12:30

Title: Simple lattices in products

Abstract: Burger and Mozes constructed the first examples of simple uniform lattices in products of trees. In this talk, we will review the key ingredients of their construction and present two related results on simple lattices. The first, joint with Ivan Levcovitz and Alex Margolis, concerns counting the number of such lattices. The second, joint with Michal Amir, introduces new examples of simple groups arising in products of trees and certain two-dimensional Davis complexes.

Maximilien Forte (UCLouvain) -- 14:00-15:00

Title: Symmetric trivalent polygonal complexes

Abstract: I will discuss J. Światkowski's work on the construction and description of symmetric trivalent polygonal complexes, a family of non-positively curved spaces closely related to cubic graph theory. We will see how properties of these complexes, such as the non-discreteness of the full automorphism group, reduce to local conditions on the links. This provides a framework for constructing new explicit examples of totally disconnected locally compact groups admitting cocompact lattices.

Stefan Witzel (JLU Giessen) -- 15:00-16:00

Title: Simple CAT(0) groups

Abstract: I speak about CAT(0) groups that are simple (like Burger-Mozes groups) but do not geometrically decompose as a direct product (unlike Burger-Mozes groups). The groups arise as lattices having the normal subgroup property (like arithmetic groups) but fail to be residually finite (unlike arithmetic groups). Although conjecturally there is a vast supply of such groups, we only know of a handful so far. This is based on joint work with Thomas Titz Mite and (independently) Jean Lécureux.

Robynn Corveleyn (UCLouvain) -- 16:30-17:30

Title: Finite quotients of Kac-Moody groups and high-dimensional expanders

Abstract: In this talk, I will discuss the construction of finite simple quotients of a generalised triangle group constructed from low rank matrix groups. I will then highlight the tools coming from the theory of Kac-Moody groups behind this construction, and illustrate how these quotients give rise to families of high-dimensional expanders. 

Ilaria Castellano (Politecnico di Torino) -- 10:00-11:00

Title: Higher Baumslag-Solitar groups and Σ-theory

Abstract: In this talk I will present the construction of a suitable cube complex X with a G-action, for G a generalized Baumslag-Solitar group BS(p_1, q_1;...;p_n, q_n) = <a, t_1,..., t_n | t_i a^{p_i} = a^{q_i} t_i, t_i t_j = t_j t_i for all i,j>. Such a complex X is isomorphic to the product of the Bass-Serre trees of the classical BS(p_i, q_i), and by devising an easy-to-check criterion for computing the essential connectivity properties of products of trees (equipped with suitable height functions), the complex can be used to describe the full Σ-theory of the generalized Baumslag-Solitar groups. This is a joint work with Kevin Klinge and José Pedro Quintanilha.

Indira Chatterji (Université Côte d'Azur) -- 11:30-12:30

Title: Strictly separated median spaces                                                                                        

Abstract: I will discuss median spaces, and in particular the ones with the property that any two points are separated by the interior of half-spaces. I will explain why this is a natural class and discuss conjectures and open problems. This is joint work with Lamine Messaci.

Sebastian Giersbach (JLU Giessen) -- 14:00-15:00

Title: Finiteness properties and quasi-isometries of graphs of tdlc groups

Abstract: Fundamental groups of finite graphs of groups behave well in regard to certain geometric properties. A theorem of Haglund-Wise relates finiteness properties of such groups to those of their vertex and edge groups. Similarly, Papasoglu-Whyte showed that under certain conditions the quasi-isometry type of such groups is determined by the quasi-isometry types of their vertex groups.

In this talk, we discuss extensions of these results to totally disconnected locally compact (tdlc) groups and apply them to a construction due to Simon Smith. Given permutation groups $M$ and $N$, the associated Smith universal group $U(M, N)$ acts on a biregular tree with local actions $M$ and $N$. Under suitable assumptions, these groups are simple non-discrete tdlc groups. Using this framework, we construct examples of simple non-discrete tdlc groups with prescribed finiteness properties. Furthermore, we obtain uncountably many pairwise non-quasi-isometric simple non-discrete compactly generated tdlc groups. Part of this talk is based on joint work with Laura Bonn.

Stephan Tornier (University of Newcastle AUS) -- 15:00-16:00

Title: Uniscalar (P)-closed groups acting on trees

Abstract: (joint work with M. Chijoff and M. Ferov). Reid-Smith recently parametrised (P)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated, discrete or simple, are reflected in its local action diagram. In this talk we outline how the scale values of a (P)-closed group can be determined from its local action diagram. As an application, we show that uniscalar (P)-closed groups are either horocyclic or compact-by-discrete, providing one possible answer to a broad question by Thomas Weigel.

Oussama Bensaid (UCLouvain) -- 09:30-10:30

Title: Coarse separation and splittings in right-angled Artin groups

Abstract: Given the Cayley graph of a finitely generated group, one can ask whether it can be separated into large connected components by removing a subset of subexponential growth. In joint work with Anthony Genevois and Romain Tessera, we investigate groups with this property. We show that the right-angled Artin groups satisfying this property are exactly those that split over an abelian subgroup, or equivalently, those whose defining graph is complete or separated by a complete subgraph.

Pénélope Azuelos (University of Bristol) -- 11:00-12:00

Title: A flat torus theorem for hierarchically hyperbolic spaces

Abstract: Hierarchically hyperbolic spaces are metric spaces which exhibit a coarse but highly organised form of nonpositive curvature. They provide a common generalisation of mapping class groups and compact special groups (e.g. right-angled Artin groups) where one can hope to adapt methods from either setting. I will present an analogue of the CAT(0) flat torus theorem for these spaces and discuss some of its consequences. This talk is based on joint work with Mark Hagen.

Ana Isakovic (University of Cambridge) -- 09:30-10:30

Title: Top-dimensional l^2-homology of hyperbolic groups

Abstract: Computing l^2-homology can be a demanding task especially if one attempts to follow the definition directly. In this talk, I will speak about using automatic structures to construct top-dimensional l^2-cycles and how they can further be used to compute l^2-homology of certain hyperbolic groups. 

Roman Sauer (KIT Karlsruhe) -- 11:00-12:00

Title: Hyperbolic Kazhdan groups with prescribed second l^2-Betti number

Abstract: We realize every positive rational number as the second l^2-Betti number of a hyperbolic Kazhdan group.
The construction relies on three ingredients:
- the theory of group-theoretic Dehn fillings
- the Cohen-Lyndon property and its excision principle
- higher property T
This is joint work with Francesco Fournier-Facio.