Fixed subgroups of automorphisms of hyperbolic 3-manifold groups

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

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