Spectrum of SL(2,R)-characters: the once-punctured torus case

TL;DR

This paper studies the spectrum of SL(2,R) representations of the once-punctured torus's fundamental group, revealing it as a Cantor set that encodes the dynamics of associated cocycles and mapping class group actions.

Contribution

It introduces the spectrum of such representations, characterizes its structure as a Cantor set for generic cases, and links it to the dynamics of cocycles and mapping class group actions.

Findings

Spectrum is a Cantor set for generic representations.

Spectrum characterizes the dynamics of associated cocycles.

Provides insights into the action of the mapping class group.

Abstract

Consider a topological surface $\Sigma$. We introduce the spectrum of a representation from the fundamental group of $\Sigma$ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.