Irreducible groups and ergodicity in the boundary
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912765670588416 |
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| author | dey, Subhadip Hurtado, Sebastian |
| author_facet | dey, Subhadip Hurtado, Sebastian |
| contents | We show that if $G$ is a real semi-simple Lie group, and $Γ$ is a discrete subgroup of $G$ containing a subgroup $Σ$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $Γ$ is not isomorphic to a free product of $Σ$ with $\mathbb{Z}$. Moreover, if $Σ$ has algebraic entries, then $Γ$ has algebraic entries as well. As a consequence, we show that if all irreducible discrete subgroups of ${\rm SL}_2(\mathbb{R}) \times {\rm SL}_2(\mathbb{R})$ act ergodically on $\mathbb{S}^1\times \mathbb{S}^1 $, such groups cannot be free groups (or even Gromov hyperbolic). In the appendix, we discuss a connection between the existence of discrete irreducible groups and diophantine properties of Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducible groups and ergodicity in the boundary dey, Subhadip Hurtado, Sebastian Group Theory Differential Geometry Dynamical Systems We show that if $G$ is a real semi-simple Lie group, and $Γ$ is a discrete subgroup of $G$ containing a subgroup $Σ$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $Γ$ is not isomorphic to a free product of $Σ$ with $\mathbb{Z}$. Moreover, if $Σ$ has algebraic entries, then $Γ$ has algebraic entries as well. As a consequence, we show that if all irreducible discrete subgroups of ${\rm SL}_2(\mathbb{R}) \times {\rm SL}_2(\mathbb{R})$ act ergodically on $\mathbb{S}^1\times \mathbb{S}^1 $, such groups cannot be free groups (or even Gromov hyperbolic). In the appendix, we discuss a connection between the existence of discrete irreducible groups and diophantine properties of Lie groups. |
| title | Irreducible groups and ergodicity in the boundary |
| topic | Group Theory Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2512.12141 |