Canonical representations of surface groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917932652560384 |
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| author | Landesman, Aaron Litt, Daniel |
| author_facet | Landesman, Aaron Litt, Daniel |
| contents | Let $Σ_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $Σ_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$ρ: π_1(Σ_{g,n})\to GL_r(\mathbb{C})$$ is a representation whose conjugacy class has finite orbit under the mapping class group, and $r<\sqrt{g+1}$, then $ρ$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz.
The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_15352 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Canonical representations of surface groups Landesman, Aaron Litt, Daniel Geometric Topology Algebraic Geometry Number Theory Representation Theory 34M56, 57K20, 14C30, 14H10, 11G99 Let $Σ_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $Σ_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$ρ: π_1(Σ_{g,n})\to GL_r(\mathbb{C})$$ is a representation whose conjugacy class has finite orbit under the mapping class group, and $r<\sqrt{g+1}$, then $ρ$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems. |
| title | Canonical representations of surface groups |
| topic | Geometric Topology Algebraic Geometry Number Theory Representation Theory 34M56, 57K20, 14C30, 14H10, 11G99 |
| url | https://arxiv.org/abs/2205.15352 |