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：

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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。