On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups
It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus $g>1$ is bounded by the linear polynomial $12(g-1)$, and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in $g$ (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension $d$, uniformizing handlebodies by Schottky groups and considering finite groups of isometries of such handlebodies. We prove that the order of a finite group of isometries of a handlebody of dimension $d$ acting faithfully on the fundamental group is bounded by a polynomial of degree $d/2$ in $g$ if $d$ is even, and of degree $(d+1)/2$ if $d$ is odd, and that the degree $d/2$ for even $d$ is best possible. This implies analogous polynomial Jordan-type bounds for arbitrary finite groups of isometries of handlebodies (since a handlebody of dimension $d>3$ admits $S^1$-actions, there does not exist an upper bound for the order of the group itself).