Valuation-like maps and the congruence subgroup property

Abstract

Let D be a finite dimensional division algebra and N a subgroup of finite index in D ×. A valuation-like map on N is a homomorphism ϕ:N?Γ from N to a (not necessarily abelian) linearly ordered group Γ satisfying N <-α+1⊆N <-α for some nonnegative α∈Γ such that N <-α≠=?, where N <-α={x∈N|ϕ(x)<-α}. We show that this implies the existence of a nontrivial valuation v of D with respect to which N is (v-adically) open. We then show that if N is normal in D × and the diameter of the commuting graph of D ×/N is ≥4, then N admits a valuation-like map. This has various implication; in particular it restricts the structure of finite quotients of D ×. 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].