Simultaneous ping-pong for finite subgroups of reductive groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911236394844160 |
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| author | Janssens, Geoffrey Temmerman, Doryan Thilmany, François |
| author_facet | Janssens, Geoffrey Temmerman, Doryan Thilmany, François |
| contents | Let $Γ$ be a Zariski-dense subgroup of a reductive group $\mathbf{G}$ defined over a field $F$. Given a finite collection of finite subgroups $H_i$ ($i \in I$) of $\mathbf{G}(F)$ avoiding the center, we establish a criterion to ensure that the set of elements of $Γ$ that form a free product with every $H_i$ (the so-called simultaneous ping-pong partners for $H_i$) is both Zariski- and profinitely dense in $Γ$. This criterion applies namely to direct products $\mathbf{G}$ of inner $\mathbb{R}$-forms of $\operatorname{(P)GL}_n$, and gives a positive answer to this particular case of a question asked by Bekka, Cowling and de la Harpe. For torsion elements, a complication arises due to the fact that a finite cyclic group can split into a direct product. When $\mathbf{G}$ is the multiplicative group of a semisimple algebra, we also give a more explicit method to obtain free products between two given finite subgroups, via first-order deformations.
In the second half, we investigate the case where $\mathbf{G}$ is the multiplicative group of the group algebra $FG$ of a finite group $G$, and $Γ$ is the group of units of an order in $FG$. In this regard, we prove that the set of bicylic units that play ping-pong with a given shifted bicyclic unit, is Zariski- and profinitely dense, addressing a long-standing belief in the field of group rings. This result is deduced from the criterion above, combined with sharp existence results for well-behaved irreducible representations of $G$ that are center-preserving on a given subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23957 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous ping-pong for finite subgroups of reductive groups Janssens, Geoffrey Temmerman, Doryan Thilmany, François Group Theory Representation Theory 20E06, 20G25, 20C05 Let $Γ$ be a Zariski-dense subgroup of a reductive group $\mathbf{G}$ defined over a field $F$. Given a finite collection of finite subgroups $H_i$ ($i \in I$) of $\mathbf{G}(F)$ avoiding the center, we establish a criterion to ensure that the set of elements of $Γ$ that form a free product with every $H_i$ (the so-called simultaneous ping-pong partners for $H_i$) is both Zariski- and profinitely dense in $Γ$. This criterion applies namely to direct products $\mathbf{G}$ of inner $\mathbb{R}$-forms of $\operatorname{(P)GL}_n$, and gives a positive answer to this particular case of a question asked by Bekka, Cowling and de la Harpe. For torsion elements, a complication arises due to the fact that a finite cyclic group can split into a direct product. When $\mathbf{G}$ is the multiplicative group of a semisimple algebra, we also give a more explicit method to obtain free products between two given finite subgroups, via first-order deformations. In the second half, we investigate the case where $\mathbf{G}$ is the multiplicative group of the group algebra $FG$ of a finite group $G$, and $Γ$ is the group of units of an order in $FG$. In this regard, we prove that the set of bicylic units that play ping-pong with a given shifted bicyclic unit, is Zariski- and profinitely dense, addressing a long-standing belief in the field of group rings. This result is deduced from the criterion above, combined with sharp existence results for well-behaved irreducible representations of $G$ that are center-preserving on a given subgroup. |
| title | Simultaneous ping-pong for finite subgroups of reductive groups |
| topic | Group Theory Representation Theory 20E06, 20G25, 20C05 |
| url | https://arxiv.org/abs/2510.23957 |