A new identity for SL(2,C)-characters of the once punctured torus group

TL;DR

This paper introduces new variations of McShane identities for SL(2,C)-representations of the once punctured torus group, focusing on those satisfying specific Bowditch conditions and invariance under Anosov mapping classes.

Contribution

It extends McShane identities to broader classes of representations and conjugacy conditions, providing new insights into the structure of these character varieties.

Findings

Derived new identities for SL(2,C)-representations

Extended identities to representations fixed by Anosov mapping classes

Enhanced understanding of Bowditch conditions in character varieties

Abstract

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.