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Andoni Zozaya

Publications and source records attributed to Andoni Zozaya.

12 recordsLinked to original sources
arXiv2026

Strong conciseness in profinite metabelian groups

A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G \in \mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Using the notion of polynomial mappings introduced by Passi, we establish that every group word is strongly concise in the class of profinite metabelian groups. With this new approach, we also give an alternative proof for the fact that every group word is strongly concise in the class of profinite nilpotent groups.

math.GR↗
arXiv2026

On the expected value of energy in groups

We obtain explicit upper and lower bounds for the expected action energy associated with a pair $({\sf A},{\sf \Delta})$ of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form $({\sf A},{\sf A})$ and $({\sf A},{\sf A}^{-1})$. Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets $|{\sf A}^{\ast 2}|$ and $|{\sf A}{\sf A}^{-1}|$.

math.GR↗
arXiv2025

Flexible $3$-valent graphs of even girth

We prove the existence of a connected flexible $3$-valent vertex-transitive graph of girth $2\ell$ for every integer $\ell$. We also give a constructive proof if $\ell$ is prime.

math.GR↗
arXiv2025

On generalisations of conciseness

Based on the notions of conciseness and semiconciseness, we show that these properties are not equivalent by proving that a word originally presented by Ol'shanskii is semiconcise but not concise. We further establish that every $1/m$-concise word is semiconcise by proving that when the group word $w$ takes finitely many values in $G$, the iterated commutator subgroup $[w(G), G, \stackrel{(m)}{\dots}, G]$ is finite for some $m \in \mathbb{N}$ if and only if $[w(G), G]$ is finite.

math.GR↗
arXiv2025

Two-generation of traceless matrices over finite fields

We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we establish curious identities that obstruct $2$-generation.

math.RA↗
arXiv2025

Strong conciseness and equationally Noetherian groups

A word $w$ is said to be concise in a class of groups if, for every $G$ in that class such that the set of $w$-values $w\{G\}$ is finite, the verbal subgroup $w(G)$ is also finite. In the context of profinite groups, the notion of strong conciseness imposes a more demanding condition on $w$, requiring that $w(G)$ is finite whenever $|w\{G\}|< 2^{\aleph_0}$. We investigate the relation between these two properties and the notion of equationally Noetherian groups, by proving that in a profinite group $G$ with a dense equationally Noetherian subgroup, $w\{G\}$ is finite whenever $|w\{G\}|< 2^{\aleph_0}$. Consequently, we conclude that every word is strongly concise in the classes of profinite linear groups, pro-$\mathcal{C}$ completions of residually $\mathcal{C}$ linear groups and pro-$\mathcal{C}$ completions of virtually abelian-by-polycyclic groups, thereby extending well-known conciseness properties of these classes of groups.

math.GR↗
arXiv2023

The degree of commutativity of wreath products with infinite cyclic top group

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups $G$: the degree of commutativity $\text{dc}_S(G)$, with respect to a given finite generating set $S$, results from considering the fractions of commuting pairs of elements in increasing balls around $1_G$ in the Cayley graph $\mathcal{C}(G,S)$. We focus on restricted wreath products the form $G = H \wr \langle t \rangle$, where $H \ne 1$ is finitely generated and the top group $\langle t \rangle$ is infinite cyclic. In accordance with a more general conjecture, we show that $\text{dc}_S(G) = 0$ for such groups $G$, regardless of the choice of $S$. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox's main auxiliary result: in `reasonably large' homomorphic images of wreath products $G$ as above, the image of the base group has density zero, with respect to certain types of generating sets.

math.GR↗
arXiv2023

A remark on Ado's Theorem for principal ideal domains

Ado's Theorem had been extended to principal ideal domains independently by Churkin and Weigel. They demonstrated that if $R$ is a principal ideal domain of characteristic zero and $\mathfrak{L}$ is a Lie algebra over $R$ which is also a free $R$-module of finite rank, then $\mathfrak{L}$ admits a finite faithful Lie algebra representation over $R$. We present a quantitative proof of this result, providing explicit bounds on the degree of the Lie algebra representations in terms of the rank of the free module. To achieve it, we generalise an established embedding theorem for complex Lie algebras: any Lie algebra as above embeds within a larger Lie algebra that decomposes as the direct sum of its nilpotent radical and another subalgebra.

math.RA↗
arXiv2021

Standard Hausdorff spectrum of compact $\mathbb{F}_p[[t]]$-analytic groups

We prove that the $\mathbb{F}_p[[t]]$-standard Hausdorff spectrum of a compact $\mathbb{F}_p[[t]]$-analytic group contains a real interval and that it coincides with the full unit interval when the group is soluble. Moreover, we show that the $\mathbb{F}_p[[t]]$-standard Hausdorff spectrum of classical Chevalley groups over $\mathbb{F}_p[[t]]$ is not full, since 1 is an isolated point thereof.

math.GR↗