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Pavel Zalesskii

Publications and source records attributed to Pavel Zalesskii.

At least 19 recordsLinked to original sources
arXiv2025

Cyclic splittings of pro-$\mathcal{C}$ groups

In this paper we proved a pro-$\mathcal{C}$ version of the Rips-Sela Theorems on splittings as an amalgamated free product or HNN-extension over a cyclic subgroup, where $\mathcal{C}$ is a class of finite groups closed for subgroups, quotients, finite direct products and extensions with abelian kernel.

math.GR↗
arXiv2025

Aritmethic lattices of $\SO(1,n)$ and units of group rings

We establish that standard arithmetic subgroups of a special orthogonal group ${\rm SO}(1,n)$ are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.

math.GR↗
arXiv2025

The profinite completion of relatively hyperbolic virtually special groups

We give a characterization of toral relatively hyperbolic virtually special groups in terms of the profinite completion. We also prove a Tits alternative for subgroups of the profinite completion $\hat G$ of a relatively hyperbolic virtually compact special group $G$ and completely describe finitely generated pro-$p$ subgroups of $\hat G$. This applies to the profinite completion of the fundamental group of a hyperbolic arithmetic manifold. We deduce that all finitely generated pro-$p$ subgroups of the congruence kernel of a standard arithmetic lattice of $SO(n,1)$ are free pro-$p$.

math.GR↗
arXiv2024

On profinite groups with the Magnus Property

A group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse-conjugate. We show that a profinite MP group $G$ is prosolvable and any quotient of it is again MP. As corollaries we obtain that the only prime divisors of $|G|$ are $2$, $3$, $5$ and $7$, and the second derived subgroup of $G$ is pronilpotent. We also show that the inverse limit of an inverse system of profinite MP groups is again MP. Finally when $G$ is finitely generated, we establish that $G$ must in fact be finite.

math.GR↗
arXiv2024

Finite subgroups of the profinite completion of good groups

Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of $G$ and those of its profinite completion $\hat{G}$. Moreover, we prove that the centralizers and normalizers in $\hat{G}$ of finite $p$-subgroups of $G$ are the closures of the respective centralizers and normalizers in $G$. With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of $G$. In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of $3$-orbifolds and of uniform standard arithmetic hyperbolic orbifolds).

math.GR↗
arXiv2024

Prosoluble subgroups of free profinite products

We describe free prosoluble subgroups of a free product of profinite groups by strengthening the theorem of Frorian Pop and answering two questions of K. Ersoy and W. Herfort. Relatively projective prosoluble groups are also described.

math.GR↗
arXiv2024

Characterizing closed curves on Riemann surfaces via homology groups of coverings

Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally determine in the homology groups of the compactifications of unramified $p$-coverings of $S$, for $p\geq 2$ a fixed prime. In particular, this gives a characterization of simple closed curves on $S$ in terms of homology groups of $p$-coverings. We then define a $p$-adic Reidemeister pairing on the fundamental group of $S$ and show that the free homotopy classes of two loops have trivial geometric intersection if and only if they are orthogonal with respect to this pairing. As an application, we give a geometric argument to prove that oriented surface groups are conjugacy $p$-separable (a combinatorial proof of this fact was recentely given by Paris).

math.AT↗
arXiv2023

Pro-$\mathcal{C}$ RAAGs

Let $\mathcal{C}$ be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-$\mathcal{C}$ group $G_Γ$ (pro-$\mathcal{C}$ RAAG for short) is the pro-$\mathcal{C}$ completion of the right-angled Artin group $G(Γ)$ associated with the finite simplicial graph $Γ$. In the first part, we describe structural properties of pro-$\mathcal{C}$ RAAGs. Among others, we describe the centraliser of an element and show that pro-$\mathcal{C}$ RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-$p$ subgroups of pro-$\mathcal{C}$ RAAGs are either free pro-$p$ or free abelian pro-$p$. In the second part, we characterise splittings of pro-$\mathcal{C}$ RAAGs in terms of the defining graph. More precisely, we prove that a pro-$\mathcal{C}$ RAAG $G_Γ$ splits as a non-trivial direct product if and only if $Γ$ is a join and it splits over an abelian pro-$\mathcal{C}$ group if and only if a connected component of $Γ$ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-$\mathcal{C}$ RAAG, in the sense of Guirardel and Levitt.

math.GR↗
arXiv2023

Cyclic splittings of pro-p groups

In this paper we prove a pro-p version of the Rips-Sela's Theorems on splittings of a group as an amalgamated free product or HNN-extension over an infinite cyclic subgroup.

math.GR↗
arXiv2023

Finitely generated normal pro-$\mathcal C$ subgroups in right angled Artin pro-$\mathcal C$ groups

Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $Γ$ be a finite simplicial graph and $G = G_Γ$ be the corresponding pro-$\mathcal C$ RAAG. We show that if $N$ is a non-trivial finitely generated, normal, full pro-$\mathcal C$ subgroup of $G$ then $G/ N$ is finite-by-abelian. In the pro-$p$ case we show a criterion for $N$ to be of type $FP_n$ when $G/ N \simeq \mathbb{Z}_p$. Furthermore for $G/ N$ infinite abelian we show that $N$ is finitely generated if and only if every normal closed subgroup $ N_0 \triangleleft G$ containing $N$ with $G/ N_0 \simeq \mathbb{Z}_p$ is finitely generated. For $G/ N$ infinite abelian with $N$ weakly discretely embedded in $G$ we show that $N$ is of type $FP_n$ if and only if every $ N_0 \leq G$ containing $N$ with $G/ N_0 \simeq \mathbb{Z}_p$ is of type $FP_n$.

math.GR↗
arXiv2023

Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation

We prove that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(\cal G, Γ)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $Γ$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $G$ to be accessible in terms of the first cohomology $H^1(G, \mathbb{F}_p[[G]])$.

math.GR↗
arXiv2020

Subgroups of pro-$p$ $\mathrm{PD}^3$-groups

We study 3-dimensional Poincaré duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal, or the subgroup is cyclic and the group is polycyclic, or the subgroup is Demushkin and normal in an open subgroup of $G$. Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincaré duality pro-$p$ groups.

math.GR↗
arXiv2020

Right-angled Artin pro-$p$ groups

Let $p$ be a prime. The right-angled Artin pro-$p$ group $G_Γ$ associated to a fnite simplicial graph $Γ$ is the pro-$p$ completion of the right-angled Artin group associated to $Γ$. We prove that the following assertions are equivalent: (i) no induced subgraph of $Γ$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of $G_Γ$ is itself a right-angled Artin pro-$p$ group (possibly infinitely generated); (iii) $G_Γ$ is a Bloch-Kato pro-$p$ group; (iv) every closed subgroup of $G_Γ$ has torsion free abelianization; (v) $G_Γ$ occurs as the maximal pro-$p$ Galois group $G_K(p)$ of some field $K$ containing a primitive $p$th root of unity; (vi) $G_Γ$ can be constructed from $\mathbb{Z}_p$ by iterating two group theoretic operations, namely, direct products with $\mathbb{Z}_p$ and free pro-$p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for right-angled Artin pro-$p$ groups. Moreover, we prove that $G_Γ$ is coherent if and only if each circuit of $Γ$ of length greater than three has a chord.

math.GR↗
arXiv2020

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation $R$-lattice for the finite $p$-group $G$ in terms of the restriction to a normal subgroup $N$ and the $N$-fixed points of the lattice, where $R$ is a finite extension of the $p$-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite $p$-group, allowing $R$ to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.

math.RT↗