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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2009.12746 |
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| _version_ | 1866914347692851200 |
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| author | Ghosh, Sourav |
| author_facet | Ghosh, Sourav |
| contents | In this article, we look at real split semisimple algebraic groups $\mathsf{G}$ with trivial center and faithful irreducible algebraic representations $\mathtt{R}$ of $\mathsf{G}$ on some vector space $\mathsf{V}$ which admit zero as a weight and which are self-contragredient (for example, adjoint representation of $\mathsf{PSL}(n,\mathbb{R})$).
We show that, there exist polynomials made out of Margulis invariants of $(g,X)\in\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$ which are also rational expressions in $(g,X)$. Moreover, we show that any Zariski dense finitely generated subgroup of $\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$, for which the linear parts of the non-identity elements are loxodromic, is isospectrally rigid with respect to the Margulis invariants. In particular, we show that Margulis--Smilga spacetimes are isospectrally rigid too. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_12746 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Isospectrality of Margulis-Smilga spacetimes for irreducible representations of real split semisimple Lie groups Ghosh, Sourav Geometric Topology In this article, we look at real split semisimple algebraic groups $\mathsf{G}$ with trivial center and faithful irreducible algebraic representations $\mathtt{R}$ of $\mathsf{G}$ on some vector space $\mathsf{V}$ which admit zero as a weight and which are self-contragredient (for example, adjoint representation of $\mathsf{PSL}(n,\mathbb{R})$). We show that, there exist polynomials made out of Margulis invariants of $(g,X)\in\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$ which are also rational expressions in $(g,X)$. Moreover, we show that any Zariski dense finitely generated subgroup of $\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$, for which the linear parts of the non-identity elements are loxodromic, is isospectrally rigid with respect to the Margulis invariants. In particular, we show that Margulis--Smilga spacetimes are isospectrally rigid too. |
| title | Isospectrality of Margulis-Smilga spacetimes for irreducible representations of real split semisimple Lie groups |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2009.12746 |