Profinite isomorphisms and fixed-point properties

Abstract

We describe a flexible construction that produces triples of finitely generated, residually finite groups $M\hookrightarrow P\hookrightarrow \Gamma$, where the maps induce isomorphisms of profinite completions $\hat{M}\cong \hat{P}\cong \hat{\Gamma }$, but $M$ and $\Gamma$ have Serre's property FA while $P$ does not. In this construction, $P$ is finitely presented and $\Gamma$ is of type ${\mathrm{F}}_{\infty }$. More generally, given any positive integer $d$, one can demand that $M$ and $\Gamma$ 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.

10 pages, no figures

References in corpus (2)

• On the finite presentation of subdirect products and the nature of residually free groups
• The homology of groups, profinite completions, and echoes of Gilbert Baumslag