• Corpus ID: 244488706

Hurwitz groups, maximal reducible groups and maximal handlebody groups

@inproceedings{Zimmermann2021HurwitzGM,
  title={Hurwitz groups, maximal reducible groups and maximal handlebody groups},
  author={Bruno Zimmermann},
  year={2021}
}

• B. Zimmermann
• Published 21 October 2021
• Mathematics

. 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. 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 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?