On Pappus and Anosov Representations of the Modular Group

TL;DR

This paper characterizes the space of certain discrete faithful representations of the modular group into isometries of a symmetric space, identifying a specific component with a clear geometric boundary and interior.

Contribution

It proves that the Barbot component of the representation space is homeomorphic to ^2 imes [0,), with the boundary representing Pappus representations and the interior Anosov representations.

Findings

The Barbot component is homeomorphic to ^2 imes [0,)

Boundary points correspond to Pappus representations

Interior points correspond to Anosov representations

Abstract

Let $X=SL_3(\R)/SO(3)$. Let $\cal DFR$ be the space of discrete faithful representations of the modular group into ${\rm Isom\/}(X)$ which map the order $2$ generator to an isometry with a unique fixed point. In this paper, we prove that $\cal DFR$ has a component $\cal B$, the so-called Barbot component, that is homeomorphic to $\R^2 \times [0,\infty)$. The boundary of $\cal B$ parametrizes the Pappus representations and the interior consists of Anosov representations.