Fixed subgroups of automorphisms of hyperbolic 3-manifold groups

For fixed subgroups Fix ( ϕ ) of automorphisms ϕ of hyperbolic 3-manifold groups π 1 ( M ) , we observe that rk ( Fix ( ϕ ) ) < 2 rk ( π 1 ( M ) ) and the constant 2 in the inequality is sharp; we also classify all possible groups Fix ( ϕ ) .