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

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.