Fixed subgroups of automorphisms of hyperbolic 3-manifold groups

Abstract

For fixed subgroups $Fix(\phi)$ of automorphisms $\phi$ on hyperbolic 3-manifold groups $\pi_{1}(M)$, we observed that $\text{rk}(Fix(\phi))<2\text{rk}(\pi_{1}(M))$ and the constant 2 in the inequality is sharp; we also classify all possible groups $Fix(\phi)$.