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 S¹-actions, there does not exist an upper bound for the order of the group itself).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm230-3-2,
author = {Mattia Mecchia and Bruno P. Zimmermann},
title = {On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups},
journal = {Fundamenta Mathematicae},
volume = {228},
year = {2015},
pages = {237-249},
zbl = {1326.57054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm230-3-2}
}
Mattia Mecchia; Bruno P. Zimmermann. On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups. Fundamenta Mathematicae, Tome 228 (2015) pp. 237-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm230-3-2/