Maximal bordered surface groups versus maximal handlebody groups

The order of a finite group of diffeomorphisms of a compact bordered surface of algebraic genus g > 1 is at most 12(g - 1), and this bound is sharp for infinitely many values of g. The same upper bound holds for the maximum possible order of a finite group of orientation-preserving diffeomorphisms of a handlebody of genus g > 1. Every finite group which acts on a compact bordered surface acts also on a handlebody of the same genus. In this paper, we show that there are infinitely many values of g for which the upper bound is attained for handlebodies but not for bordered surfaces. Quite remarkably, for 1 < g < 2000 the only such value is g = 161