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Commensurability growths of algebraic groups
Mathematische Zeitschrift ( IF 1.0 ) Pub Date : 2019-05-14 , DOI: 10.1007/s00209-019-02334-5
Khalid Bou-Rabee , Tasho Kaletha , Daniel Studenmund

Fixing a subgroup $$\varGamma $$ Γ in a group G , the full commensurability growth function assigns to each n the cardinality of the set of subgroups $$\varDelta $$ Δ of G with $$[\varGamma : \varGamma \cap \varDelta ][\varDelta : \varGamma \cap \varDelta ] \le n$$ [ Γ : Γ ∩ Δ ] [ Δ : Γ ∩ Δ ] ≤ n . For pairs $$\varGamma \le G$$ Γ ≤ G , where G is a higher rank Chevalley group scheme defined over $${\mathbb {Z}}$$ Z and $$\varGamma $$ Γ is an arithmetic lattice in G , we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on G .

中文翻译:

代数群的可公度增长

将子群 $$\varGamma $$Γ 固定在群 G 中,完全可公度增长函数将 G 的子群集合 $$\varDelta $$ Δ 的基数分配给每个 n,其中 $$[\varGamma : \varGamma \ cap \varDelta ][\varDelta : \varGamma \cap \varDelta ] \le n$$ [ Γ : Γ ∩ Δ ] [ Δ : Γ ∩ Δ ] ≤ n 。对于 $$\varGamma \le G$$ Γ ≤ G 对,其中 G 是在 $${\mathbb {Z}}$$ Z 上定义的更高阶 Chevalley 群方案,$$\varGamma $$ Γ 是算术格在 G 中,我们给出了完全可公度增长的精确估计,将其与子组增长和仅取决于 G 的可计算不变量相关联。
更新日期:2019-05-14
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