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## Fundamenta Mathematicae

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

### Tom 230 / 2015

Fundamenta Mathematicae 230 (2015), 237-249 MSC: 57S17, 57S25, 57N16. DOI: 10.4064/fm230-3-2

#### Streszczenie

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).

#### Autorzy

• Mattia MecchiaDipartimento di Matematica e Geoscienze
Università degli Studi di Trieste
34127 Trieste, Italy
e-mail
• Bruno P. ZimmermannDipartimento di Matematica e Geoscienze
Università degli Studi di Trieste
34127 Trieste, Italy
e-mail

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