Cohomological and geometric invariants of simple complexes of groups

Petrosyan, Nansen; Prytuła, Tomasz

doi:10.2140/agt.2024.24.1121

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Cohomological and geometric invariants of simple complexes of groups

Nansen Petrosyan and Tomasz Prytuła

Algebraic & Geometric Topology 24 (2024) 1121–1155

DOI: 10.2140/agt.2024.24.1121

Abstract

We investigate cohomological properties of fundamental groups of strictly developable simple complexes of groups $X$ . We obtain a polyhedral complex equivariantly homotopy equivalent to $X$ of the lowest possible dimension. As applications, we obtain a simple formula for proper cohomological dimension of $CAT (0)$ groups whose actions admit a strict fundamental domain; for any building of type $(W, S)$ that admits a chamber transitive action by a discrete group, we give a realisation of the building of the lowest possible dimension equal to the virtual cohomological dimension of $W$ ; under general assumptions, we confirm a folklore conjecture on the equality of Bredon geometric and cohomological dimensions in dimension one; finally, we give a new family of counterexamples to the strong form of Brown’s conjecture on the equality of virtual cohomological dimension and Bredon cohomological dimension for proper actions.

Keywords

complex of groups, classifying space, standard development, Coxeter system, building, virtual cohomological dimension, Bredon cohomological dimension

Mathematical Subject Classification

Primary: 05E18, 05E45, 20F65

Secondary: 20E08, 20J06

References

Publication

Received: 3 March 2022

Revised: 25 July 2022

Accepted: 1 September 2022

Published: 12 April 2024

Authors

Nansen Petrosyan
School of Mathematical Sciences University of Southampton Southampton United Kingdom

Tomasz Prytuła
Department Of Applied Mathematics And Computer Science Technical University of Denmark Lyngby Denmark

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Volume 24, issue 2 (2024)