Valuation-like maps and the congruence subgroup property

Let D be a finite dimensional division algebra and N a subgroup of finite index in D^x. 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 + I ⊆ 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 ν of D with respect to which N is (ν-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 ≥ 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].