The fixed subgroups of homeomorphisms of Seifert manifolds

Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold B_M, and f_π: π₁(M) → π₁(M) be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of f_π, namely, rankFix(f_π) < 2rankπ₁(M), which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.