Property FA is not a profinite property

TL;DR

This paper demonstrates that Property FA is not a profinite property by constructing infinitely many pairs of finitely presented, residually finite groups with isomorphic profinite completions where one has Property FA and the other does not.

Contribution

It provides the first explicit examples showing Property FA is not detectable from the profinite completion of a group.

Findings

Existence of infinitely many such pairs of groups

Property FA is not a profinite property

Counterexamples to previous assumptions

Abstract

We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups $\Delta$ and $\Gamma$ with $\Delta$ having Property FA, $\Gamma$ having a non-trivial action on a tree and $\Delta$ and $\Gamma$ having isomorphic profinite completions.