Equivariant cohomology, lattices, and trees

Hughes, Sam (2021) Equivariant cohomology, lattices, and trees. University of Southampton, Doctoral Thesis, 167pp.

Record type: Thesis (Doctoral)

Abstract

This is a ‘three paper thesis’, the main body of which consists of the following papers:
[1] S. Hughes, Cohomology of Fuchsian groups and non-Euclidean crystallographic groups, preprint, available at arXiv:1910.00519 [math.GR], 2019.
[2] S. Hughes, On the equivariant K- and KO-homology of some special linear groups, to appear in Algebraic and Geometric topology. Available at arXiv:2004.08199 [math.KT], 2020.
[3] I. Chatterji, S. Hughes and P. Kropholler, Groups acting on trees and the first l²-Betti number, to appear in Proceedings of the Edinburgh Mathematical Society. Available at arXiv:2004.08199 [math.GR], 2020.
[4] S. Hughes, Graphs and complexes of lattices, preprint, available at arXiv:2104.13728 [math.GR], 2021.
[5] S. Hughes, Hierarchically hyperbolic groups, products of CAT (-1) spaces, and virtual torsion-freeness, preprint, available at arXiv:2105.02847 [math.GR], 2021.

In [1], we compute the cohomology groups of a number of low dimensional linear groups. In particular, for each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.

In [2], we study K-theoretic properties of arithmetic groups in relation to the Baum–Connes Conjecture. Specifically, we compute the equivariant KO-homology of the classifying space for proper actions of SL₃(Z), and the Bredon homology and equivariant K-homology of the classifying spaces for proper actions of SL₂(Z[1/p]) for each prime p. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture on positive scalar curvature for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.

In [3], we generalise results of Thomas, Allcock, Thom-Petersen, and Kar-Niblo to the first l²-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.

In [4], we study lattices acting on CAT(0) spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of CAT(0) lattices. Using this framework we characterise irreducible uniform (Isom(Eⁿ) x T)-lattices by C*-simplicity and the failure of virtual fibring and biautomaticity. We construct non-residually finite uniform lattices acting on arbitrary products of right angled buildings and non-biautomatic lattices acting on the product of E ⁿ and a right-angled building. We investigate the residual finiteness, L²-cohomology, and C*-simplicity of CAT lattices more generally. Along the way we prove that many right angled Artin groups with rank 2 centre are not quasi-isometrically rigid.

In [5], we prove that a group acting geometrically on a product of proper minimal CAT(-1) spaces without permuting isometric factors is a hierarchically hyperbolic group. As an application we construct, what to the author’s knowledge are, the first examples of hierarchically hyperbolic groups which are not virtually torsion-free.