# On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

## TL;DR

This paper characterizes 3-manifolds and finite groups that achieve the maximal order of orientation-reversing actions on Heegaard splittings, linking geometric structures to group actions.

## Contribution

It provides a classification of 3-manifolds and groups realizing the maximal finite group-action order on Heegaard splittings, including reducible and irreducible cases.

## Key findings

- Maximal order of group actions is 24(g-1) for genus g > 1.
- Reducible manifolds are doubles of handlebodies with maximal actions.
- Irreducible manifolds are quotients of geometries by Coxeter groups.

## Abstract

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1), and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order |G| = 24(g-1) is obtained, for some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it is obtained by doubling an action of maximal possible order 24(g-1) on a handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.