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 then
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 ).Comment: 13 pages; this is the final version to appear in Fund. Mat
