Cohomology of uniserial 𝑝-adic space groups

Abstract

A decade ago, J. F. Carlson proved that there are finitely many cohomology rings of finite 2 2 -groups of fixed coclass, and he conjectured that this result ought to be true for odd primes. In this paper, we prove the non-twisted case of Carlson’s conjecture for any prime and we show how to proceed in the twisted case.

Cohomology of p-groups of nilpotency class smaller than p

“…In the same paper, Carlson conjectures that the analogous result should hold for the p odd case, that is, he conjectures that there are finitely many isomorphism types of cohomology algebras for all p-groups of fixed coclass. This result has been partially proved first by Eick and Green in [4] and later by the authors of this manuscript in [3]. In the former paper, Eick and Green prove that there are finitely many Quillen categories of p-groups of fixed coclass m. In the latter paper, Díaz Ramos, Garaialde Ocaña and González-Sánchez prove Carlson's conjecture for the non-twisted p-groups of coclass m [3, Theorem 7.1].…”

“…In this section, we shall prove some counting argument results using spectral sequences that will be essential to prove the main result. Actually, the following results can be considered as a generalization of [3,Lemma 4.3 and Theorem 4.4].…”

Cohomology of uniserial p-adic space groups with cyclic point group

“…In fact, for all i ∈ Z the semidirect product θ ⋉ T i is also the unique maximal class pro-p group and for each n ≥ 1, T −n /T 0 ⋊ θ is the unique quotient of T 0 ⋊ C p of size p n+1 . Now, [2,Proposition 5.8] gives the following result. It is readily checked that η p x commutes with θ and thus, η p x G. Now, by [6, Theorem 7.4.2] and [7, Lemma 8.1], R is just infinite.…”

“…By abusing the notation, let Λ * (Y ) denote both the exterior algebra and the differential graded algebra equipped with the zero differential. There is a morphism of differential graded modules Page 7] or in [2, Section 5.1], that induces an isomorphism in the ideal of nilpotent elements of the cohomology algebra H * (K 0 ; F p ) (see [2,Lemma 5.3]). Following Notation 1 above and for all i ≥ 0, let ϕ i :…”

“…In the same paper [1], Carlson also conjectures that an analogous result should hold for odd primes and he claims that one of the key steps is to prove that all the quotients of the unique pro-p group of maximal class have isomorphic cohomology algebras [1,Question 6.1]. In [2], the authors prove that there are finitely many algebras realizing such cohomology groups [2,Proposition 5.7] while in [4], G. Ellis proves that certain periodically constructed quotients of crystalographic groups have isomorphic mod-p cohomology groups.…”