On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups

TL;DR

This paper generalizes bounds on the size of finite groups of isometries acting on handlebodies to arbitrary dimensions, establishing polynomial bounds depending on the dimension and genus, and confirming the optimality of these bounds.

Contribution

It extends known results from 3- and 4-dimensional handlebodies to all dimensions, providing explicit polynomial bounds on group orders based on dimension and genus.

Findings

Order of finite isometry groups bounded by polynomial in genus

Bounds depend on dimension, degree d/2 for even, (d+1)/2 for odd

Optimality of the polynomial degree for even dimensions

Abstract

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 ).