Abstract

Let M be a compact Kleinian 3-manifold with non-empty and connected boundary S. Then S carries the structure of a closed Riemann surface and by symmetries of S or M we understand their antiholomorphic involutions. In this paper we provide upper bounds for the number of conjugacy classes of symmetries of M in terms of the genus of S. Furthermore, we show that our bounds are sharp for hyperbolic handlebodies for infinitely many genera, by explicit constructions of finite normal extensions of certain Schottky groups, using at decisive stage of construction, quasi-conformal deformation theory of Riemann surfaces and Teichmuller theory of Fuchsian groups. In particular, we obtain that a Kleinian. 3-manifold of even genus g has at most four nonconjugate symmetries and that this bound is achieved for arbitrary even g. Motivated by the behavior of Riemann surfaces, we propose the problem of the validity of our bounds, obtained for hyperbolic handlebodies, in a purely topological setting. Finally, since handlebodies can be seen as fattening-up of graphs, the results of our paper may be interpreted in terms of certain automorphisms of graphs and we propose another problem concerning it at the end of the paper.