Let Γ be a discrete group which is a split extension of a group Δ by a Coxeter group W, with Δ acting on W by Coxeter graph automorphisms with kernel Δ₀. Let M_i, i = 1,2, be two Γ-manifolds (possibly with boundary) such that the isotropy groups are finite and the fixed point sets are contractible and W acts by reflections. Let f be a Γ-homotopy equivalence between them that it is a homeomorphism outside the orbit of a compact subset. Then f is Γ-homotopic to a Γ-homeomorphism, provided that certain finite extensions of Δ₀ that fix the faces of the fundamental domains are topologically rigid groups.

The first author was partially supported by Hungarian Nat. Found. for Sci. Research Grant T032478.