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

Mattia Mecchia, and Bruno P. Zimmermann. "On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups." Fundamenta Mathematicae 230.3 (2015): 237-249. <http://eudml.org/doc/283396>.

@article{MattiaMecchia2015,
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 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).},
author = {Mattia Mecchia, Bruno P. Zimmermann},
journal = {Fundamenta Mathematicae},
keywords = {handlebody; finite group action; Schottky group; Jordan-type bound},
language = {eng},
number = {3},
pages = {237-249},
title = {On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups},
url = {http://eudml.org/doc/283396},
volume = {230},
year = {2015},
}

TY - JOUR
AU - Mattia Mecchia
AU - Bruno P. Zimmermann
TI - On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups
JO - Fundamenta Mathematicae
PY - 2015
VL - 230
IS - 3
SP - 237
EP - 249
AB - 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).
LA - eng
KW - handlebody; finite group action; Schottky group; Jordan-type bound
UR - http://eudml.org/doc/283396
ER -