Pro-𝑝 link groups and 𝑝-homology groups

Abstract

We characterize the groups of branched twist spins of classical knots in terms of 3-manifold groups, and also give a purely algebraic, conjectural characterization in terms of P D 3 -groups. We show also that each group is the group of at most finitely many branched twist spins.

Nilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag

“…One says that a group Γ is pro-p good if for each n ≥ 0, the homomorphism of cohomology groups H n ( Γ p ; F p ) → H n (Γ; F p ) induced by the natural map Γ → Γ p is an isomorphism, where the group on the left is in the continuous cohomology of Γ p . One says that the group Γ is cohomologically complete if Γ is pro-p good for all primes p. It is shown in [12] that many link groups are cohomologically complete, and indeed it was claimed by Hillman, Matei and Morishita [34] that all link groups are cohomologically complete. Counterexamples to the method of [34] were given in [11], but [11] left open the possibility that link groups might nevertheless be cohomologically complete.…”

On pro-𝑝 link groups of number fields

“…Regarding ∆(1 + T ) ∈ Λ, Iwasawa invariants λ, µ and ν for X cyc /X are analogously defined and studied (cf. [15,18,19,20,34,50,51] etc.). For L = K 1 ∪ K 2 ⊂ M = S 3 (and X cyc such that the meridians of K 1 and K 2 are nontrivial in Γ), an analogue [19, Theorem 3.2] of Gold's theorem states that ∆(1 + T )Λ = ΛT if and only if the linking number lk(K 1 , K 2 ) ≡ 0 (mod p).…”

-adic Mahler measure and -covers of links

“…Finally we remark that a recent development of such a study for twisted Alexander polynomials of knots is due to Tange ([Tan17]). The following formula is an analogue of Iwasawa's class number formula ([Iwa59]), which was initially given by [HMM06] and generalized by [KM08] and [Uek17]:…”