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We consider large finite group-actions on surfaces and discuss and comparevarious notions for such actions: Hurwitz actions and Hurwitz groups; maximalreducible and completely reducible actions; bounding and geometrically boundingactions; maximal handlebody groups and maximal bounded surface groups; inparticular, we discuss small simple groups of various types.  A Hurwitz group is a finite group of orientation-preserving diffeomorphismsof 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-preservingdiffeomorphisms of maximal possible order $12(g-1)$ of a 3-dimensionalhandlebody of genus $g>1$. Among others, we consider the question of when aHurwitz group acting on a surface of genus $g$ contains a subgroup of maximalpossible order $12(g-1)$ extending to a handlebody (or, more generally, amaximal reducible group extending to a product with handles), and show thatsuch Hurwitz groups are closely related to the smallest Hurwitz group ${\rmPSL}_2(7)$ of order 168 acting on Klein's quartic of genus 3. We discuss simplegroups of small order which are maximal handlebody groups and, more generaly,maximal reducible groups. We discuss also the problem of which Hurwitz actionsbound geometrically, and in particular whether Klein's quartic boundsgeometrically: does there exist a compact hyperbolic 3-manifold with totallygeodesic boundary isometric to Klein's quartic? Finally, large bounding andnon-bounding actions on surfaces of genus 2, 3 and 4 are discussed in section3.

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