Equivariant bounded cohomology

Abstract

This research paper thesis consists of the following articles:
[1] K. Li. Bounded cohomology of classifying spaces for families of subgroups.
To appear in Algebr. Geom. Topol., arXiv:2105.05223, 2021.
[2] K. Li, C. Löh, and M. Moraschini. Bounded acyclicity and relative simplicial
volume. Preprint, arXiv:2202.05606, 2022.
[3] K. Li. Amenable covers of right-angled Artin groups.
Preprint, arXiv:2204.01162, 2022.
[4] K. Li. On the topological complexity of toral relatively hyperbolic groups.
Proc. Amer. Math. Soc., 150(3):967–974, 2022.
In [1], we introduce a bounded version of Bredon cohomology for groups relative to a
family of subgroups. We obtain cohomological characterisations of relative amenability
and relative hyperbolicity, analogous to the results of Johnson and Mineyev for bounded
cohomology.
In [2], joint with Clara Löh and Marco Moraschini, we provide new vanishing and glueing
results for relative simplicial volume. We consider equivariant nerve pairs and relative
classifying spaces for families of subgroups. Our methods also lead to vanishing results
for `2-Betti numbers of aspherical CW-pairs with small relative amenable category.
In [3], we consider the right-angled Artin group AL associated to a finite flag complex L.
We show that the amenable category of AL equals the virtual cohomological dimension
of the right-angled Coxeter group WL.
In [4], we prove that the topological complexity TC() equals cd( ) for certain toral
relatively hyperbolic groups .

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Identifiers

ORCID for Nansen Petrosyan: orcid.org/0000-0002-2768-5279

ORCID for Ian Leary: orcid.org/0000-0001-8300-4979

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