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Pavel Zalesskii

Publications and source records attributed to Pavel Zalesskii.

At least 37 records · Page 2Linked to original sources
arXiv2019

Profinite groups in which centralizers are virtually procyclic

The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.

math.GR↗
arXiv2019

Retracts of free groups and a question of Bergman

Let $F_n$ be a free group of finite rank $n \geq 2$. We prove that if $H$ is a subgroup of $F_n$ with $\textrm{rk}(H)=2$ and $R$ is a retract of $F_n$, then $H \cap R$ is a retract of $H$. However, for every $m \geq 3$ and every $1 \leq k \leq n-1$, there exist a subgroup $H$ of $F_n$ of rank $m$ and a retract $R$ of $F_n$ of rank $k$ such that $H \cap R$ is not a retract of $H$. This gives a complete answer to a question of Bergman. Furthermore, we provide positive evidence for the inertia conjecture of Dicks and Ventura. More precisely, we prove that $\textrm{rk}(H \cap \textrm{Fix}(S)) \leq \textrm{rk}(H)$ for every family $S$ of endomorphisms of $F_n$ and every subgroup $H$ of $F_n$ with $\textrm{rk}(H) \leq 3$.

math.GR↗
arXiv2018

Infinitely Generated virtually free pro-$p$ groups and $p$-adic representations

We prove the pro-$p$ version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro-$p$ group $G$ has finite centralizes of all non-trivial torsion elements more stronger statement is proved: $G$ embeds into a free pro-$p$ product of a free pro-$p$ group and finite $p$-group. The integral $p$-adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro-$p$ case.

math.GR↗
arXiv2018

Profinite detection of 3-manifold decompositions

The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.

math.GR↗
arXiv2018

Profinite groups in which centralizers are abelian

The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.

math.GR↗
arXiv2017

Profinite extensions of centralizers and the profinite completion of limit groups

We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.

math.GR↗
arXiv2017

Pro-$p$ subgroups of profinite completions of 3-manifold groups

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.

math.GR↗
arXiv2017

A restricted Magnus property for profinite surface groups

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for ${\mathscr S}$ a class of finite groups, we prove that, if $x$ and $y$ are \emph{algebraically simple} elements of the pro-${\mathscr S}$ completion $\hatΠ^{\mathscr S}$ of an orientable surface group $Π$, such that, for all $n\in{\mathbb N}$, there holds $\langle x^n\rangle^{\hatΠ^{\mathscr S}}=\langle y^n\rangle^{\hatΠ^{\mathscr S}}$, then $x$ is conjugated to $y^s$ for some $s\in(\hat{\mathbb Z}^{\mathscr S})^\ast$. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions. The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists to profinite Dehn multitwists.

math.GR↗
arXiv2017

Virtual retraction and Howson's theorem in pro-$p$ groups

We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $τ\colon U \to K$ satisfying $τ(k) = k$ for every $k \in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products, and deduce that Howson's theorem holds for the Sylow subgroups of the absolute Galois group of a number field. Finally, we confirm two conjectures of Ribes, thus classifying the finitely generated pro-$p$ M. Hall groups.

math.GR↗
arXiv2016

Distinguishing geometries using finite quotients

We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $\hat{\mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-$p$ subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-$p$.

math.GR↗
arXiv2016

Virtually compact special hyperbolic groups are conjugacy separable

We prove that any word hyperbolic group which is virtually compact special (in the sense of Haglund and Wise) is conjugacy separable. As a consequence we deduce that all word hyperbolic Coxeter groups and many classical small cancellation groups are conjugacy separable. To get the main result we establish a new criterion for showing that elements of prime order are conjugacy distinguished. This criterion is of independent interest; its proof is based on a combination of discrete and profinite (co)homology theories.

math.GR↗
arXiv2016

Virtually free pro-p products

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).

math.GR↗
arXiv2015

Conjugacy Distinguished Subgroups

Let ${\cal C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually ${\cal C}$ group $R$ is said to be conjugacy ${\cal C}$-distinguished if whenever $y\in R$, then $y$ has a conjugate in $H$ if and only if the same holds for the images of $y$ and $H$ in every quotient group $R/N\in {\cal C}$ of $R$. We prove that in a group having a normal free subgroup $Φ$ such that $R/Φ$ is in ${\cal C}$, every finitely generated subgroup is conjugacy ${\cal C}$-distinguished. We also prove that finitely generated subgroups of limit groups, of Lyndon groups and certain one-relator groups are conjugacy distinguished (${\cal C}$ here is the class of all finite groups).

math.GR↗
arXiv2013

Normal subgroups of fundamental groups of affine curves in positive characteristic II

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $π_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of $π_1(C)$. In "Normal subgroups of fundamental groups of affine curves in positive characteristic" we proved the same result under the additional hypothesis that $k$ had countable cardinality.

math.AG↗
arXiv2013

Subgroups and Homology of Extensions of Centralizers of Pro-p Groups

We study asymptotic behavior of the dimensions of the homology groups of subgroups of finite index in finitely generated subgroups of pro-$p$ extension of centralizers of free pro-$p$ groups. We also prove group theoretic structure properties of such pro-$p$ groups that are of independent interest.

math.GR↗
arXiv2013

Splitting theorems for pro-$p$ groups acting on pro-$p$ trees and 2-generated subgroups of free pro-$p$ products with procyclic amalgamations

Let G be a finitely generated infinite pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. Then we prove that G splits as a pro-p amalgamated product or as a pro-p HNN-extension over an edge stabilizer. Using this result we prove under certain conditions that free pro-p products with procyclic amalgamation inherit from its free factors the property of each 2-generated subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogs of classical results in abstract combinatorial group theory.

math.GR↗
arXiv2012

One-relator groups with torsion are conjugacy separable

We prove that one-relator groups with torsion are hereditarily conjugacy separable. Our argument is based on a combination of recent results of Dani Wise and the first author. As a corollary we obtain that any quasiconvex subgroup of a one-relator group with torsion is also conjugacy separable.

math.GR↗