Bounds for fixed points on hyperbolic manifolds

For a compact (without boundary) hyperbolic n-manifold M with n≥4, we show that there exists a finite bound B such that for any homeomorphism f:M→M and any fixed point class F of f, the index |ind(f,F)|≤B, which is a partial positive answer of a question given by Jiang in [3]. Moreover, when M is a compact hyperbolic 4-manifold, or a compact hyperbolic n-manifold (n≥5) with isometry group Isom(M) a p-group, we give some explicit descriptions of the bound B.