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

Mattia Mecchia ¹   ; Bruno P. Zimmermann ¹

¹ Dipartimento di Matematica e Geoscienze Università degli Studi di Trieste 34127 Trieste, Italy

Fundamenta Mathematicae, Tome 230 (2015) no. 3, pp. 237-249

Résumé

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).

Détail

DOI : 10.4064/fm230-3-2

Keywords: known order finite group diffeomorphisms dimensional handlebody genus bounded linear polynomial g order finite group diffeomorphisms dimensional handlebody equivalently its boundary manifold faithful fundamental group bounded quadratic polynomial linear present paper prove generalization handlebodies arbitrary dimension uniformizing handlebodies schottky groups considering finite groups isometries handlebodies prove order finite group isometries handlebody dimension acting faithfully fundamental group bounded polynomial degree even degree odd degree even best possible implies analogous polynomial jordan type bounds arbitrary finite groups isometries handlebodies since handlebody dimension admits actions there does exist upper bound order group itself