  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
$\widehat{M}\cong\widehat{P}\cong\widehat{\Gamma}$, 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 ${\rm{F}}_\infty$. 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.
