1,721,041 research outputs found
Hierarchically hyperbolic groups, products of spaces, and virtual torsion-freeness
No full textWe prove that a group acting geometrically on a product of proper minimal spaces without permuting isometric factors is a hierarchically hyperbolic group. As an application we construct hierarchically hyperbolic groups which are not virtually torsion-free
Cohomology of Fuchsian groups and non-Euclidean crystallographic groups
No full textFor each geometrically finite 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. Finally, we compute the -Betti numbers of the NEC groups
Graphs and complexes of lattices
No full textWe study lattices acting on 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 lattices. Using this framework we characterise irreducible uniform -lattices by -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 and a right-angled building. We investigate the residual finiteness, -cohomology, and -simplicity of lattices more generally. Along the way we prove that many right angled Artin groups with rank centre are not quasi-isometrically rigid
On the equivariant K- and KO-homology of some special linear groups
No full textWe compute the equivariant -homology of the classifying space for proper actions of and . We also compute the Bredon homology and equivariant -homology of the classifying spaces for proper actions of and for each prime . Finally, we prove the unstable Gromov-Lawson-Rosenberg conjecture for when
Equivariant cohomology, lattices, and trees
No full textThis 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
The character table of a sharply 5-transitive subgroup of the alternating group of degree 12
Get PDFWe calculate the character table of a sharply 5-transitive subgroup of Alt(12), and of a sharply 4-transitive subgroup of Alt(11). Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.</p
Homological growth of Artin kernels in positive characteristic
No full textWe prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina–Brady groups. Specifically, we prove that the mod p homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.</p
The first ℓ<sup>2</sup>-Betti number and groups acting on trees
No full textWe 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
Coherence for elementary amenable groups
No full textWe prove that for an elementary amenable group, coherence of the group, homological coherence of the group, and coherence of the group ring are all equivalent. This generalizes a result of Bieri and Strebel for finitely generated soluble groups
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