Hurwitz groups, maximal reducible groups and maximal handlebody groups
Abstract
A Hurwitz group is a finite group of orientationpreserving diffeomorphisms of maximal possible order $84(g1)$ of a closed orientable surface of genus $g>1$. A maximal handlebody group instead is a group of orientationpreserving diffeomorphisms of maximal possible order $12(g1)$ of a 3dimensional handlebody of genus $g>1$. We consider the question of when a Hurwitz group acting on a surface of genus $g$ contains a subgroup of maximal possible order $12(g1)$ 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 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 3manifold with totally geodesic boundary isometric to Klein's quartic?
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.11050
 Bibcode:
 2021arXiv211011050Z
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Group Theory
 EPrint:
 12 pages