The fixed subgroups of homeomorphisms of Seifert manifolds

TL;DR

This paper establishes a bound on the rank of fixed subgroups of automorphisms induced by orientation-reversing homeomorphisms of Seifert manifolds, extending known inequalities from surface and hyperbolic 3-manifold groups.

Contribution

It provides a new inequality bounding the fixed subgroup rank of automorphisms in Seifert manifolds, linking topological properties to algebraic automorphism behavior.

Findings

Bound on fixed subgroup rank: < 2 * rank of fundamental group

Extension of inequalities from surface and hyperbolic 3-manifold groups

Applicable to automorphisms induced by orientation-reversing homeomorphisms

Abstract

Let $M$ be a compact connected orientable Seifert manifold with hyperbolic orbifold $B_M$, and $f_{\pi}: \pi_1(M)\rightarrow\pi_1(M)$ be an automorphism induced by an orientation-reversing homeomorphism $f$ of $M$. We give a bound on the rank of the fixed subgroup of $f_{\pi}$, namely, $\rank\fix(f_{\pi})<2\rank \pi_1(M)$, which is similar to the inequalities on surface groups and hyperbolic 3-manifold groups.