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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

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.

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
Received: 3 March 2022
Revised: 25 July 2022
Accepted: 1 September 2022
Published: 12 April 2024
Nansen Petrosyan
School of Mathematical Sciences
University of Southampton
United Kingdom
Tomasz Prytuła
Department Of Applied Mathematics And Computer Science
Technical University of Denmark

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