Profinite isomorphisms and fixed-point properties

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