Deformation of Fuchsian representations and proper affine actions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916312884707328 |
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| author | Ghosh, Sourav |
| author_facet | Ghosh, Sourav |
| contents | The main goal of this article is to generalize Mess' work and using results from Labourie--Wentworth, Potrie--Sambarino and Smilga, to show that inside Hitchin representations, infinitesimal deformations of Fuchsian representations of a cocompact surface group do not act properly along the directions corresponding to the sum of a mixed odd differential and a $2m$-differential for any $1\leq m \leq \lfloor\frac{n}{2}\rfloor$.
In the process, we introduce affine versions of cross ratios and triple ratios. We introduce Margulis invariants and relate them with affine crossratios and infinitesimal Jordan projections. We obtain a general equivalent criterion for existence of proper affine actions in terms of the structure of the Margulis invariant spectra. Also, using a stability argument we show the existence of proper affine actions of non-abelian free groups whose linear part is a Hitchin representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16655 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Deformation of Fuchsian representations and proper affine actions Ghosh, Sourav Geometric Topology Differential Geometry 53-xx, 37-xx The main goal of this article is to generalize Mess' work and using results from Labourie--Wentworth, Potrie--Sambarino and Smilga, to show that inside Hitchin representations, infinitesimal deformations of Fuchsian representations of a cocompact surface group do not act properly along the directions corresponding to the sum of a mixed odd differential and a $2m$-differential for any $1\leq m \leq \lfloor\frac{n}{2}\rfloor$. In the process, we introduce affine versions of cross ratios and triple ratios. We introduce Margulis invariants and relate them with affine crossratios and infinitesimal Jordan projections. We obtain a general equivalent criterion for existence of proper affine actions in terms of the structure of the Margulis invariant spectra. Also, using a stability argument we show the existence of proper affine actions of non-abelian free groups whose linear part is a Hitchin representation. |
| title | Deformation of Fuchsian representations and proper affine actions |
| topic | Geometric Topology Differential Geometry 53-xx, 37-xx |
| url | https://arxiv.org/abs/2312.16655 |