Albert Garreta Fontelles

University of the Basque Country | UPV/EHU · Departamento de Matemáticas

In this paper, we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by...

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. In particular, these simple groups have infinite w-verbal width for all proper words w.

In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by...

We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family F of one-relator monoids of the form 〈A|w=1〉 where for each monoid M in F, the longstanding open problem of decidability of word equations with length constraints reduces to the Diophantine problem (i.e. decidability of systems of equations)...

We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring $R$ we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implie...

We study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2 , then the...

In this paper, we study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, that is, coincides with the positive theory of a non-abelian free group. Our results apply to a wide class of groups, including non-virtually solvable fundamental gr...

We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that if $|R| \leq |A|-2$ then the Diophantine proble...

We study finitely generated nilpotent groups G given by full rank finite presentations 〈A|R〉Nc in the variety Nc of nilpotent groups of class at most c, where c≥2. We prove that if the deficiency |A|−|R| is at least 2 then the group G is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almos...

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring o...

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. In particular, these simple groups have infinite $w$-verbal width for all non-trivial words $w$.

We study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, i.e. coincides with the positive theory of a non-abelian free group. Our results apply to a wide class of groups, including non-virtually solvable fundamental groups of $3$-manifol...

Let p be a prime and let G be a subgroup of a Sylow pro-p subgroup of the group of automorphisms of the p-adic tree. We prove that if G is fractal and \(|G':{{\mathrm{st}}}_G(1)'|=\infty \), then the set L(G) of left Engel elements of G is trivial. This result applies to fractal nonabelian groups with torsion-free abelianization, for example the Ba...

We investigate systems of equations and the first-order theory of one-relator monoids and of word-hyperbolic monoids. We describe a family of one-relator monoids of the form $\langle A\mid w=1\rangle$ with decidable Diophantine problem (i.e.\ decidable systems of equations), and another family $\mathcal{F}$ of one-relator monoids $\langle A\mid w=1...

We study systems of equations in different classes of solvable groups. For each group $G$ in one of these classes we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by systems of equations (e-interpretable). This leads to the conjecture that $\mathbb{Z}$ is e-interpretable in $G$ and that the Diophantine proble...

We study systems of equations in different families of rings and algebras. In each such structure $R$ we interpret by systems of equations (e-interpret) a ring of integers $O$ of a global field. The long standing conjecture that $\mathbb{Z}$ is always e-interpretable in $O$ then carries over to $R$, and if true it implies that the Diophantine probl...

An element $g$ of a given group $G$ is a left Engel element if for every $x \in G$ there exists an integer $n = n(g, x) \geq 1$ such that $[x, g, \overset{n}{\dots}, g] = 1$. The set of such elements is denoted by $L(G)$. We prove that $L(G)=1$ for a certain class of groups that are weakly regular branch over their derived subgroup, and that have t...

We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\tau_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \label{tau2pres_0}\langle A, C \mid [a_i, a_j]= \prod_t {\scriptstyle c_t^{\scriptscriptstyle \lambda_{t,i,j}}} \ (i< j), \ [A,C]=[C,C]=1\rangle,$ whe...

We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\tau_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \label{tau2pres_0}\langle A, C \mid [a_i, a_j]= \prod_t {\scriptstyle c_t^{\scriptscriptstyle \lambda_{t,i,j}}} \ (i< j), \ [A,C]=[C,C]=1\rangle,$ whe...

We study random nilpotent groups of the form $G=N/\langle\langle R \rangle \rangle$, where $N$ is a non-abelian free nilpotent group with $m$ generators, and $R$ is a set of $r$ random relators of length $\ell$. We prove that the following holds asymptotically almost surely as $\ell\to \infty$: 1) If $r\leq m-2$, then the ring of integers $\mathbb{...