THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

The weight of nonstrongly complete profinite groups

We compute the local weight of the completion of a nonstrongly complete profinite group and conclude that, if a profinite group is abstractly isomorphic to its own profinite completion, then they are equal. The local weights of all the groups in the tower of completions are computed as well.

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

Finiteness of Isomorphism Classes of Hyperbolic Polycurves with Prescribed Fundamental Groups

In the present paper, we show that there are at most finitely many isomorphism classes of hyperbolic polycurves (i.e., successive extensions of families of hyperbolic curves) over certain types of fields whose étale fundamental group is isomorphic to a prescribed profinite group.

Profinite groups in which the probabilistic zeta function has no negative coefficients

To a finitely generated profinite group [Formula: see text], a formal Dirichlet series [Formula: see text] is associated, where [Formula: see text] and [Formula: see text] denotes the Möbius function of the lattice of open subgroups of [Formula: see text] Its formal inverse [Formula: see text] is the probabilistic zeta function of [Formula: see text]. When [Formula: see text] is prosoluble, every coefficient of [Formula: see text] is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

Compact groups with many elements of bounded order

Lévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

The virtually generating graph of a profinite group

We consider the graph $\Gamma _{\text {virt}}(G)$ whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

Drinfeld’s lemma for diamonds

This chapter addresses Drinfeld's lemma for diamonds. It proves a local analogue of Drinfeld's lemma, thereby giving a first nontrivial argument involving diamonds. This lecture is entirely about fundamental groups. A diamond is defined to be connected if it is not the disjoint union of two open subsheaves. For a connected diamond, finite étale covers form a Galois category. As such, for a geometric point, one can define a profinite group, such that finite sets are equivalent to finite étale covers. In this proof, the chapter uses the formalism of diamonds rather heavily to transport finite étale maps between different presentations of a diamond as the diamond of an analytic adic space.