McShane’s identity for classical Schottky groups

Ser Peow Tan, Yan Loi Wong and Ying Zhang

Vol. 237 (2008), No. 1, 183–200

DOI: 10.2140/pjm.2008.237.183

Abstract

In 1998, Greg McShane demonstrated a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. In 2006, we generalized this to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical Schottky groups. As a consequence, we obtain some surprising new identities in the case of Fuchsian Schottky groups. For classical Schottky groups of rank 2, we also give generalizations of the Weierstrass identities, given by McShane in 2004.

Keywords

Schottky group, McShane identity, Fuchsian group, simple closed curve, Weierstrass point, one-holed torus, three-holed sphere

Mathematical Subject Classification 2000

Primary: 32G15

Secondary: 30F60, 57M50

Milestones

Received: 13 December 2007

Revised: 5 May 2008

Accepted: 13 May 2008

Published: 1 September 2008