Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910209978400768 |
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| author | Schwartz, Richard Evan |
| author_facet | Schwartz, Richard Evan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group Schwartz, Richard Evan Geometric Topology 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. |
| title | Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2412.18457 |