Valuation-like maps and the congruence subgroup property

• 成果类型: Article
• 署名作者: Rapinchuk, AS; Segev, Y
• 署名单位: University of Virginia; Ben-Gurion University of the Negev
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s002220100136
• 发表日期: 2001
• 页码: 571-607
• 关键词: division algebra; multiplicative subgroups; finite index; ring

摘要

Let D be a finite dimensional division algebra and N a subgroup of finite index in DX. A valuation-like map on N is a homomorphism psi: N --> T from N to a (not necessarily abelian) linearly ordered group T satisfying N<-alpha + 1 subset of or equal to N<-alpha for some nonnegative alpha is an element of T such that N<-alpha not equal 0 where N<-alpha = {x is an element of N / psi (x) < -alpha}. We show that this implies the existence of a nontrivial valuation v of D with respect to which N is (nu -adically) open. We then show that if N is normal in D-x and the diameter of the commuting graph of D-X/N is greater than or equal to 4, then N admits a valuation-like map. This has various implication; in particular it restricts the structure of finite quotients of D X. The notion of a valuation-like map is inspired by [27], and in fact is closely related to part (U3) of the U-Hypothesis in [27].