A note on Reidemeister Torsion of G-Anosov Representations

TL;DR

This paper proves the well-definedness of Reidemeister torsion for G-Anosov representations of surface groups and derives a new formula relating it to the Atiyah-Bott-Goldman symplectic form, with applications to Hitchin components.

Contribution

It establishes the well-definedness of Reidemeister torsion for G-Anosov representations and introduces a novel formula linking it to the symplectic form of the Lie group G.

Findings

Reidemeister torsion is well-defined for G-Anosov representations.

A new formula relates R-torsion to the Atiyah-Bott-Goldman symplectic form.

Applications to Hitchin components and Teichmüller space are demonstrated.

Abstract

This article considers $G$-Anosov representations of a fixed closed oriented Riemann surface $\Sigma$ of genus at least $2$. Here, $G$ is the Lie group $\text{PSp}(2n,\mathbb{R}$), $\text{PSO}(n,n)$ or $\text{PSO}(n,n+1)$. It proves that Reidemeister torsion (R-torsion) associated to $\Sigma$ with coefficients in the adjoint bundle representations of such representations is well-defined. Moreover, by using symplectic chain complex method, it establishes a novel formula for R-torsion of such representations in terms of the Atiyah-Bott-Goldman symplectic form corresponding to the Lie group $G$. Furtermore, it applies the results to Hitchin components, in particular, Teichm\"{u}ller space.