Planar graphs with separation are dp-minimal
We prove that given a planar embedding of a graph in the sphere the expansion of the graph structure by predicates encoding separation of vertices by simple cycles of the graph is dp-minimal.
math.CO↗
arXiv subjects
Javier de la Nuez González
Publications and source records attributed to Javier de la Nuez González.
Group topologies on automorphism groups of homogeneous structures
We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.
math.LO↗
On the elementary theory of graph products of groups
In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition which generalizes a range of natural ``non-freeness conditions" such as the satisfaction of a group law, having nontrivial center or being boundedly simple. As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups to describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results on the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent or classical linear groups.
math.GR↗
Simple groups with infinite verbal width and the same positive theory as free groups
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 $w$.
math.GR↗
On the positive theory of groups acting on trees
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 $3$-manifold groups, generalised Baumslag-Solitar groups and almost all one-relator groups and graph products of groups. It follows that groups in the class satisfy a number of algebraic properties: for instance, their verbal subgroups have infinite width and, although some groups in the class are simple, they cannot be boundedly simple. In order to prove these results we describe a uniform way for constructing (weak) small cancellation tuples from (weakly) stable elements. This result of interest in its own is fundamental to obtain corollaries of general nature such as a quantifier reduction for positive sentences or the preservation of the non-trivial positive theory under extensions of groups.
math.GR↗
Linearity of graph products
In this work we prove that, given a simplicial graph $Γ$ and a family $\mathcal{G}$ of linear groups over a domain $R$, the graph product $Γ\mathcal{G}$ is linear over $R[\underline t]$, where $\underline t$ is a tuple of finitely many linearly independent variables. As a consequence we obtain that any graph product of finitely many groups linear over the complex numbers is again a linear group over the complex numbers. This solves an open problem of Hsu and Wise in the case of faithful representations over $\mathbb C$.
math.GR↗
Some lower bounds on Shelah rank in the free group
We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.
math.LO↗
On expansions of non-abelian free groups by cosets of a finite index subgroup
Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarskiś problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov's theorem on $\forall\exists$-sentences and show that the positive theory depends only on $Q$ and neither on the rank of $F$ nor the particular quotient map.
math.GR↗