arXiv subjects
Publications and source records attributed to Thomas S. Weigel.
Using methods of associative algebras, Lie theory, group cohomology, and modular representation theory, we construct profinite $p$-adic analytic groups such that the centralizer of each of their non-trivial elements is abelian. The paper answers questions of P.~Shumyatsky, P.~Zalesskii, and T.~Zapata in the Israel J. Math., v.~230, 2019.
For a prime number $\ell$ we introduce and study oriented right-angled Artin pro-$\ell$ groups $G_{Γ,λ}$(oriented pro-$\ell$ RAAGs for short) associated to a finite oriented graph $Γ$ and a continuous group homomorphism $λ\colon\mathbb Z_\ell\to\mathbb Z_\ell^\times$. We show that an oriented pro-$\ell$ RAAG $G_{Γ,λ}$ is a Bloch-Kato pro-$\ell$ group if, and only if, $(G_{Γ,λ},θ_{Γ,λ})$ is an oriented pro-$\ell$ group of elementary type generalizing a recent result of I. Snopche and P. Zalesskii. Here $θ_{Γ,λ}\colon G_{Γ,λ}\to\mathbb Z_p^\times$ denotes the canonical $\ell$-orientation on $G_{Γ,λ}$. We invest some effort in order to show that oriented right-angled Artin pro-$\ell$ groups share many properties with right-angled Artin pro-$\ell$-groups or even discrete RAAG's, e.g., if $Γ$ is a specially oriented chordal graph, then $G_{Γ,λ}$ is coherent, generalizing a result of C. Droms. Moreover, in this case $(G_{Γ,λ},θ_{Γ,λ})$ has the Positselski-Bogomolov property generalizing a result of H. Servatius, C. Droms and B. Servatius for discrete RAAG's. If $Γ$ is a specially oriented chordal graph and ${\rm Im}(λ)\subseteq 1+4\mathbb Z_2$ in case that $\ell=2$, then ${\rm H}^\bullet(G_{Γ,λ},\mathbb F_\ell) \simeq Λ^\bullet(\ddotΓ^{\rm op})$ generalizing a well known result of M. Salvetti.
For a prime number $\ell$ we say that an oriented pro-$\ell$ group $(G,θ)$ has the Bogomolov-Positselski property if the kernel of the canonical projection on its maximal $θ$-abelian quotient $π^{ab}_{G,θ}\colon G\to G(θ)$ is a free pro-$\ell$ group contained in the Frattini subgroup of $G$. We show that oriented pro-$\ell$ groups of elementary type have the Bogomolov-Positselski property. This shows that Efrat's Elementary Type Conjecture implies a positive answer to Positselski's version of Bogomolov's Conjecture on maximal pro-$\ell$ Galois groups of a field $K$ in case that $K^\times/(K^\times)^\ell$ is finite. Secondly, it is shown that for an $H^\bullet$-quadratic oriented pro-$\ell$ group $(G,θ)$ the Bogomolov-Positselski property can be expressed by the injectivity of the transgression map $d_2^{2,1}$ in the Hochschild-Serre spectral sequence.
Motivated by their study of pro-p limit groups, D.H. Kochloukova and P.A. Zalesskii formulated a question concerning the minimum number of generators d(N) of a normal subgroup N of prime index p in a non-abelian limit group G (cf. Question*). It is shown that the analogous question for the rational rank has an affirmative answer (cf. Thm. A). From this result one may conclude that the original question of D.H. Kochloukova and P.A. Zalesskii has an affirmative answer if the abelianization G^{\ab} of G is torsion free and d(G)=d(G^{\ab}) (cf. Cor.~B), or if G has the IF-property (cf. Thm. C).
A finite group $G$ is called {\it $p^i$-central of height $k$} if every element of order $p^i$ of $G$ is contained in the $k^{th}$-term $ζ_k(G)$ of the ascending central series of $G$. If $p$ is odd such a group has to be $p$-nilpotent (Thm. A). Finite $p$-central $p$-groups of height $p-2$ can be seen as the dual analogue of finite potent $p$-groups, i.e., for such a finite $p$-group $P$ the group $P/Ω_1(P)$ is also $p$-central of height $p-2$ (Thm. B). In such a group $P$ the index of $P^p$ is less or equal than the order of the subgroup $Ω_1(P)$ (Thm. C). If the Sylow $p$-subgroup $P$ of a finite group $G$ is $p$-central of height $p-1$, $p$ odd, and $N_G(P)$ is $p$-nilpotent, then $G$ is also $p$-nilpotent (Thm. D). Moreover, if $G$ is a $p$-soluble finite group, $p$ odd, and $P\in \text{Syl}_p(G)$ is $p$-central of height $p-2$, then $N_G(P)$ controls $p$-fusion in $G$ (Thm. E). It is well-known that the last two properties hold for Swan groups.