Large finite group actions on surfaces: Hurwitz groups, maximal reducible and maximal handlebody groups, bounding and non-bounding actions

TL;DR

This paper explores various large finite group actions on surfaces, comparing different notions such as Hurwitz groups, handlebody groups, and bounding actions, with a focus on their maximal properties and specific small simple groups.

Contribution

It provides a detailed comparison of different large finite group actions on surfaces, introduces new relationships between Hurwitz groups and handlebody groups, and examines bounding properties of actions on low-genus surfaces.

Findings

Hurwitz groups of maximal order relate closely to small simple groups like PSL_2(7).

Certain Hurwitz groups contain subgroups extending to handlebodies of maximal order.

Existence of hyperbolic 3-manifolds with Klein's quartic as totally geodesic boundary is discussed.

Abstract

We consider large finite group-actions on surfaces and discuss and compare various notions for such actions: Hurwitz actions and Hurwitz groups; maximal reducible and completely reducible actions; bounding and geometrically bounding actions; maximal handlebody groups and maximal bounded surface groups; in particular, we discuss small simple groups of various types. A Hurwitz group is a finite group of orientation-preserving diffeomorphisms of maximal possible order $84(g-1)$ of a closed orientable surface of genus $g>1$. A maximal handlebody group instead is a group of orientation-preserving diffeomorphisms of maximal possible order $12(g-1)$ of a 3-dimensional handlebody of genus $g>1$. Among others, we consider the question of when a Hurwitz group acting on a surface of genus $g$ contains a subgroup of maximal possible order $12(g-1)$ extending to a handlebody (or, more generally, a maximal reducible group extending to a product with handles), and show that such Hurwitz groups are closely related to the smallest Hurwitz group ${\rm PSL}_2(7)$ of order 168 acting on Klein's quartic of genus 3. We discuss simple groups of small order which are maximal handlebody groups and, more generaly, maximal reducible groups. We discuss also the problem of which Hurwitz actions bound geometrically, and in particular whether Klein's quartic bounds geometrically: does there exist a compact hyperbolic 3-manifold with totally geodesic boundary isometric to Klein's quartic? Finally, large bounding and non-bounding actions on surfaces of genus 2, 3 and 4 are discussed in section 3.