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

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.