Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group

TL;DR

This paper studies the structure of a specific space of discrete faithful representations of the modular group into isometries of a symmetric space, revealing its topology, boundary, and geometric properties related to geodesic patterns and Anosov representations.

Contribution

It characterizes the Barbot component of the representation space, describing its topology, boundary parametrization, and the geometric and shearing structures of its members.

Findings

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

Boundary points correspond to Pappus representations

Interior points extend known Anosov representations with geometric patterns

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. I prove many things about the component $\cal B$ of $\cal DFR$ known as the Barbot component: It is homeomorphic to $\R^2 \times [0,\infty)$. The boundary parametrizes the Pappus representations from [{\bf S0\/}]. The interior parametrizes the complete extension of the family of Anosov representations from [{\bf BLV\/}]. The members of $\cal B$ are isometry groups of embedded patterns of geodesics in $X$ which have asymptotic properties like the edges of the Farey triangulation or shears thereof. The Anosov representations are obtained from the Pappus representations by either of two shearing operations in $X$. The shearing structure is encoded by two proper foliations of $\cal B$ into rays.