Profinite isomorphisms and fixed-point properties

Bridson, Martin R

doi:10.2140/agt.2024.24.4103

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Profinite isomorphisms and fixed-point properties

Martin R Bridson

Algebraic & Geometric Topology 24 (2024) 4103–4114

DOI: 10.2140/agt.2024.24.4103

Abstract

We describe a flexible construction that produces triples of finitely generated, residually finite groups $M ↪ P ↪ Γ$ , where the maps induce isomorphisms of profinite completions $\hat{M} ≅ \hat{P} ≅ \hat{Γ}$ , but $M$ and $Γ$ have Serre’s property FA while $P$ does not. In this construction, $P$ is finitely presented and $Γ$ is of type $F_{\infty}$ . More generally, given any positive integer $d$ , one can demand that $M$ and $Γ$ have a fixed point whenever they act by semisimple isometries on a complete CAT $(0)$ space of dimension at most $d$ , while $P$ acts without a fixed point on a tree.

Keywords

Grothendieck pairs, profinite properties, CAT(0) spaces, property FA

Mathematical Subject Classification

Primary: 20E18, 20F67, 20J05

Secondary: 20E08

References

Publication

Received: 22 June 2023

Revised: 9 September 2023

Accepted: 1 October 2023

Published: 9 December 2024

Authors

Martin R Bridson
Mathematical Institute University of Oxford Oxford United Kingdom

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Volume 24, issue 7 (2024)