Groups with infinitely many types of fixed subgroups

It is a theorem of Shor that ifG is a word-hyperbolic group, then up to isomrphism, only finitely many groups appear as fixed subgroups of automorphisms ofG. We give an example of a groupG acting freely and cocompactly on a CAT(0) square complex such that infinitely many non-isomorphic groups appear as fixed subgroups of automorphisms ofG. Consequently, Shor’s finiteness result does not hold if the negative curvature condition is relaxed to either biautomaticity or nonpositive curvature.