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In this paper we study the Roe index of the signature operator of manifoldsof bounded geometry. Our main result is the proof of the uniform homotopyinvariance of this index. In other words we show that, given anorientation-preserving uniform homotopy equivalence $f:(M,g) \longrightarrow(N,h)$ between two oriented manifolds of bounded geometry, we have that$f_\star(Ind_{Roe}D_M) = Ind_{Roe}(D_N)$. Moreover we also show that the sameresult holds considering a group $\Gamma$ acting on $M$ and $N$ by isometriesand assuming that $f$ is $\Gamma$-equivariant. The only assumption on theaction of $\Gamma$ is that the quotients are again manifolds of boundedgeometry.

Uniform homotopy invariance of Roe Index of the signature operator