McShane's Identity in Rank One Symmetric Spaces

TL;DR

This paper explores McShane's identity within real and complex hyperbolic spaces, extending it to representations of surface groups into isometry groups of rank one symmetric spaces, and unifies various proof techniques.

Contribution

It generalizes McShane's identity to broader settings and introduces a unified approach for proving such identities in rank one symmetric spaces.

Findings

Generalized McShane's identity for real and complex hyperbolic spaces

Unified proof methods for identities in rank one symmetric spaces

Extended identities to representations of surface groups

Abstract

In this paper we study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this class of identities.