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Fixed points of imaginary reflections on hyperbolic handlebodies

Published online by Cambridge University Press:  28 September 2009

RUBEN A. HIDALGO  and
BERNARD MASKIT
RUBEN A. HIDALGO
Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile. e-mail: ruben.hidalgo@usm.cl
BERNARD MASKIT
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook NY 11794-3651, U.S.A. e-mail: bernie@math.sunysb.edu
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Abstract

A Klein–Schottky group is an extended Kleinian group, containing no reflections and whose orientation-preserving half is a Schottky group. A dihedral-Klein–Schottky group is an extended Kleinian group generated by two different Klein–Schottky groups, both with the same orientation-preserving half. We provide a structural description of the dihedral-Klein–Schottky groups.

Let M be a handlebody of genus g, with a Schottky structure. An imaginary reflection τ of M is an orientation-reversing homeomorphism of M, of order two, whose restriction to its interior is an hyperbolic isometry having at most isolated fixed points. It is known that the number of fixed points of τ is at most g + 1; τ is called a maximal imaginary reflection if it has g + 1 fixed points. As a consequence of the structural description of the dihedral-Klein–Schottky groups, we are able to provide upper bounds for the cardinality of the set of fixed points of two or three different imaginary reflections acting on a handlebody with a Schottky structure. In particular, we show that maximal imaginary reflections are unique.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 135 - 158
Copyright
Copyright © Cambridge Philosophical Society 2009

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