Uniform homotopy invariance of Roe Index of the signature operator

In this paper we study the Roe index of signature operator manifolds bounded geometry. Our main result is proof uniform homotopy invariance index. other words show that, given an orientation-preserving equivalence $$f:(M,g) \longrightarrow (N,h)$$ between two oriented geometry, $$f_\star (Ind_{Roe}D_M) = Ind_{Roe}(D_N)$$ . Moreover also that same holds if a group $$\Gamma $$ acts on M and N by isometries f -equivariant. The only assumption action quotients are again