Profinite isomorphisms and fixed-point properties - Bridson, M (Mathematical Sciences Publishers, 2024)
We describe a flexible construction that produces triples of finitely generated, residually finite groups M ãÑ P ãÑ Γ, where the maps induce isomorphisms of profinite completions Mx – Pp – Γ, but p M and Γ have Serre’s property FA while P does not. In this construction, P is finitely presented and Γ is of type F8. 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 CATp0q space of dimension at most d, while P acts without a fixed point on a tree.