arXiv subjects

Ilir Snopce

Publications and source records attributed to Ilir Snopce.

15 recordsLinked to original sources

arXiv2023

On the self-similarity of the norm one group of $p$-adic division algebras

Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of $SL_1(D)$ admits self-similar actions on regular rooted trees. Moreover, we prove results on $\mathbb{Z}_p$-Lie lattices that allow to deal with the case where the center of $D$ is bigger than $\mathbb{Q}_p$, and lead to the classification of the torsion-free $p$-adic analytic pro-$p$ groups $G$ of dimension less than $p$ with the property that all the nontrivial closed subgroups of $G$ admit a self-similar action on a $p$-ary tree. As a consequence, we obtain that a nontrivial torsion-free $p$-adic analytic pro-$p$ group $G$ of dimension less than $p$ is isomorphic to the maximal pro-$p$ Galois group of a field that contains a primitive $p$-th root of unity if and only if all the nontrivial closed subgroups of $G$ admit a self-similar action on a regular rooted $p$-ary tree.

math.GR↗

arXiv2022

Frattini-resistant direct products of pro-$p$ groups

A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto Φ(H)$, from the poset of all closed subgroups of $G$ into itself, is a poset embedding. Frattini-resistant pro-$p$ groups appear naturally in Galois theory. Indeed, every maximal pro-$p$ Galois group over a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is strongly Frattini-resistant. Let $G_1$ and $G_2$ be non-trivial pro-$p$ groups. We prove that $G_1 \times G_2$ is strongly Frattini-resistant if and only if one of the direct factors $G_1$ or $G_2$ is torsion-free abelian and the other one has the property that all of its closed subgroups have torsion-free abelianization. As a corollary we obtain a group theoretic proof of a result of Koenigsmann on maximal pro-$p$ Galois groups that admit a non-trivial decomposition as a direct product. In addition, we give an example of a group that is not strongly Frattini-resistant, but has the property that its Frattini-function defines an order self-embedding of the poset of all topologically finitely generated subgroups.

math.GR↗

arXiv2022

On self-similarity of $p$-adic analytic pro-$p$ groups of small dimension

Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with $\mathrm{dim}(G) < p$, we show that the self-similar actions of $G$ on regular rooted trees can be studied through the virtual endomorphisms of the associated $\mathbb{Z}_p$-Lie lattice. We explicitly classify 3-dimensional unsolvable $\mathbb{Z}_p$-Lie lattices for $p$ odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free $p$-adic analytic pro-$p$ groups for $p\geqslant 5$, and to determine which of them admit a faithful self-similar action on a $p$-ary tree. In particular, we show that no open subgroup of $SL_1^1(Δ_p)$ admits such an action. On the other hand, we prove that all the open subgroups of $SL_2^{\triangle}(\mathbb{Z}_p)$ admit faithful self-similar actions on regular rooted trees.

math.GR↗

arXiv2020

On the self-similarity index of $p$-adic analytic pro-$p$ groups

Let $p$ be a prime. We say that a pro-$p$ group is self-similar of index $p^k$ if it admits a faithful self-similar action on a $p^k$-ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-$p$ group $G$ is defined to be the least power of $p$, say $p^k$, such that $G$ is self-similar of index $p^k$. We show that for every prime $p\geqslant 3$ and all integers $d$ there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-$p$ groups of self-similarity index greater than $d$. This implies that, in general, for self-similar $p$-adic analytic pro-$p$ groups one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

math.GR↗

arXiv2020

Frattini-injectivity and Maximal pro-$p$ Galois groups

We call a pro-$p$ group $G$ Frattini-injective if distinct finitely generated subgroups of $G$ have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-$p$ groups (and several other related concepts). Most notably, we classify the $p$-adic analytic and the solvable Frattini-injective pro-$p$ groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-$p$ group. We prove that every maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is Frattini-injective. In addition, we show that many substantial results on maximal pro-$p$ Galois groups are in fact consequences of Frattini-injectivity. For instance, a $p$-adic analytic or solvable pro-$p$ group is Frattini-injective if and only if it can be realized as a maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$); and every Frattini-injective pro-$p$ group contains a unique maximal abelian normal subgroup.

math.GR↗

arXiv2020

Index-stable compact $p$-adic analytic groups

A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let $p$ be a prime, and let $G$ be a compact $p$-adic analytic group with associated $\mathbb{Q}_p$-Lie algebra $\mathcal{L}(G)$. We prove that $G$ is index-stable whenever $\mathcal{L}(G)$ is semisimple. In particular, a just-infinite compact $p$-adic analytic group is index-stable if and only if it is not virtually abelian. Within the category of compact $p$-adic analytic groups, this gives a positive answer to a question of C. Reid. In the Appendix, J-P. Serre proves that $G$ is index-stable if and only if the determinant of any automorphism of $\mathcal{L}(G)$ has $p$-adic norm 1.

math.GR↗

arXiv2020

On pro-$p$ groups with quadratic cohomology

The main purpose of this article is to study pro-$p$ groups with quadratic $\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-quadratic pro-$p$ groups. Prime examples of such groups are the maximal Galois pro-$p$ groups of fields containing a primitive root of unity of order $p$. We show that the amalgamated free product and HNN-extension of $H^\bullet$-quadratic pro-$p$ groups is $H^\bullet$-quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-$p$ groups that yields many new examples of $H^\bullet$-quadratic groups: $p$-RAAGs. These examples generalise right angled Artin groups in the category of pro-$p$ groups. Finally, we explore "Tits alternative behaviour" of $H^\bullet$-quadratic 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

On hereditarily self-similar $p$-adic analytic pro-$p$ groups

A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of dimension less than $p$ that are strongly hereditarily self-similar of index $p$. Moreover, we show that a solvable torsion-free $p$-adic analytic pro-$p$ group of dimension less than $p$ is strongly hereditarily self-similar of index $p$ if and only if it is isomorphic to the maximal pro-$p$ Galois group of some field that contains a primitive $p$-th root of unity. As a key step for the proof of the above results, we classify the 3-dimensional solvable torsion-free $p$-adic analytic pro-$p$ groups that admit a faithful self-similar action on a $p$-ary tree, completing the classification of the 3-dimensional torsion-free $p$-adic analytic pro-$p$ groups that admit such actions.

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↗

arXiv2015

Asymptotic density of test elements in free groups and surface groups

An element $g$ of a group $G$ is a test element if every endomorphism of $G$ that fixes $g$ is an automorphism. Let $G$ be a free group of finite rank, an orientable surface group of genus $n \geq 2$, or a non-orientable surface group of genus $n \geq 3$. Let $\mathcal{T}$ be the set of test elements of $G$. We prove that $\mathcal{T}$ is a net. From this result we derive that $\mathcal{T}$ has positive asymptotic density in $G$. This answers a question of Kapovich, Rivin, Schupp, and Shpilrain. Furthermore, we prove that $\mathcal{T}$ is dense in the profinite topology on $G$.

math.GR↗

arXiv2015

Test elements in pro-$p$ groups with applications in discrete groups

Let $G$ be a group. An element $g \in G$ is called a test element of $G$ if for every endomorphism $φ:G \to G$, $φ(g)=g$ implies that $φ$ is an automorphism. We prove that for a finitely generated profinite group $G$, $g \in G$ is a test element of $G$ if and only if it is not contained in a proper retract of $G$. Using this result we prove that an endomorphism of a free pro-$p$ group of finite rank which preserves an automorphic orbit of a non-trivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-$p$ groups and Demushkin groups. By relating test elements in finitely generated residually finite-$p$ Turner groups to test elements in their pro-$p$ completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.

math.GR↗

arXiv2015

Uncountably many non-commensurable finitely presented pro-$p$ groups

Let $m\geq 3$ be a positive integer. We prove that there are uncountably many non-commensurable metabelian uniform pro-$p$ groups of dimension $m$. Consequently, there are uncountably many non-commensurable finitely presented pro-$p$ groups with minimal number of generators $m$ (and minimal number of relations $ {m \choose 2}$).

math.GR↗

arXiv2012

A characterisation of uniform pro-p groups

Let p be a prime. Uniform pro-p groups play a central role in the theory of p-adic Lie groups. Indeed, a topological group admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup which is uniform. Furthermore, uniform pro-p groups naturally correspond to powerful Lie lattices over the p-adic integers and thus constitute a cornerstone of p-adic Lie theory. In the present paper we propose and supply evidence for the following conjecture, aimed at characterising uniform pro-p groups. Suppose that p > 2 and let G be a torsion-free pro-p group of finite rank. Then G is uniform if and only if its minimal number of generators is equal to the dimension of G as a p-adic manifold, i.e., d(G) = dim(G). In particular, we prove that the assertion is true whenever G is soluble or p > dim(G).

math.GR↗

arXiv2012

Subgroup properties of pro-p extensions of centralizers

We prove that a finitely generated pro-$p$ group acting on a pro-$p$ tree $T$ with procyclic edge stabilizers is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively, in the following two situations: 1) the action is $n$-acylindrical, i.e., any non-identity element fixes not more than $n$ edges; 2) the group $G$ is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro-$p$ groups from the class $\mathcal{L}$ defined and studied in [Math. Z. 267 (2011), 109-128] as pro-$p$ analogues of limit groups. We prove that every pro-$p$ group $G$ from the class $\mathcal{L}$ is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups with infinite procyclic or trivial edge groups and finitely generated vertex groups; moreover, all non-abelian vertex groups are from the class $\mathcal{L}$ of lower level than $G$ with respect to the natural hierarchy. This allows us to give an affirmative answer to questions 9.1 and 9.3 in [Math. Z. 267 (2011), 109-128]. Namely, we prove that a group $G$ from the class $\mathcal{L}$ has Euler-Poincaré characteristic zero if and only if it is abelian, and if every abelian pro-$p$ subgroup of $G$ is procyclic and $G$ itself is not procyclic, then $def(G) \geq 2$. Moreover, we prove that $G$ satisfies the Greenberg-Stallings property and any finitely generated non-abelian subgroup of $G$ has finite index in its commensurator. We also show that all non-solvable Demushkin groups satisfy the Greenberg-Stallings property and each of their finitely generated non-trivial subgroups has finite index in its commensurator.

math.GR↗