A series associated to generating pairs of a once punctured torus group and a proof of McShane's identity
type: Item
instance of: scholarly article
title: A series associated to generating pairs of a once punctured torus group and a proof of McShane's identity
zbMATH Open document ID: 1233.57011
author: Toshihiro Nakanishi
published in: Hiroshima Mathematical Journal
publication date: 2011-10-05T00:00:00Z
review text: In his paper [Bull. Lond. Math. Soc. 36, No. 2, 181--187 (2004; Zbl 1052.57021)], \textit{G. McShane} proved that for any finite-area hyperbolic metric on a punctured torus, one has the following identity: \[ \sum_{\gamma}\arcsin \left(\frac{1}{\cosh(|\gamma|/2)}\right)=\pi/2 \] where the sum is taken over all simple closed geodesics passing through a fixed pair of Weierstrass points. In the paper under review, the author provides a new proof of this identity based on an investigation of the arrangement of axes of simple hyperbolic elements in a once-punctured torus-group which are represented by palindromic words. The author's argument also includes a short proof of the fact that the linear measure of the infinitesimal Birman-Series set is zero.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 5953595
zbMATH Keywords: simple closed geodesics
zbMATH Keywords: McShane identity
zbMATH Keywords: hyperbolic geometry
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1301586287
OpenAlex ID: W1508988511
DOI: 10.32917/HMJ/1301586287
description: scientific article; zbMATH DE number 5953595