Research papers
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- Quotient-saturated groups (with M. Roy and E. Ventura).
Journal of Pure and Applied Algebra , 229(9), pp. 108053. (2025).We introduce the new notion of quotient-saturation as a measure of the immensity of the quotient structure of a group, and we prove that it is a necessary condition for a non-elementary finitely presented group to embed in a hyperbolic group. More generally, we present a sufficient condition — called Congruence Extension Property equipment (in short, CEP-equipment) — for a finitely presented group to be quotient-saturated. Using this property, we deduce that non-elementary finitely presented subgroups of a hyperbolic group (in particular, non-elementary hyperbolic groups themselves) are quotient-saturated. Finally, we elaborate on the previous results to extend the scope of CEP-equipment (and hence of quotient-saturation) to finitely presented acylindrically hyperbolic groups. - Free-abelian by free groups: homomorphisms and algorithmic explorations (with A. Carvalho).
Kyoto Journal of Mathematics , (2025). (2025).We obtain an explicit description of the endomorphisms of free-abelian by free groups together with a characterization of when they are injective and surjective. As a consequence we see that free-abelian by free groups are Hopfian and not coHopfian, and we investigate the isomorphism problem and the Brinkmann Problem for this family of groups. In particular, we prove that the isomorphism problem (undecidable in general) is decidable when restricted to finite actions, and that the Brinkmann Problem is decidable both for monomorphisms and automorphisms. - Decidability of the Brinkmann Problems for endomorphisms of the free group (with A. Carvalho).
Archiv der Mathematik , 123(3), pp. 233-240. (2024).Building on the work of Brinkmann and Logan, we show that both the Brinkmann problem and the Brinkmann conjugacy problem are decidable for endomorphisms of the free group \(\mathbb{F}_n\). - Intersection configurations in free and free times free-abelian groups (with M. Roy and E. Ventura).
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , 154(5), pp. 1552-1582. (2024).In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group \(G\) if there exist subgroups \(H_1,\ldots,H_k \leqslant G\) realizing it. It is well known that free groups \(F_n\) satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups \(F_n \times \mathbb{Z}^m\) are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given \(k\geqslant 2\) finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a basis for it. We finally prove that any intersection configuration is realizable in an FTFA group \(F_n \times \mathbb{Z}^m\), for \(n\geqslant 2\) and large enough \(m\). As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable. -
- Stallings automata for free-times-abelian groups: intersections and index (with E. Ventura).
Publicacions Matemàtiques , 66(02), pp. 789-830. (2022).We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively. -
- A list of applications of Stallings automata (with E. Ventura).
Transactions on Combinatorics , 11(3), pp. 181-235. (2022).This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case. - On the lattice of subgroups of a free group: complements and rank (with P. V. Silva).
journal of Groups, Complexity, Cryptology , 12(1), pp. 1-24. (2020).A \(\vee\)-complement of a subgroup \(H \leqslant \mathbb{F}_n\) is a subgroup \(K \leqslant \mathbb{F}_n\) such that \(H \vee K = \mathbb{F}_n\). If we also ask \(K\) to have trivial intersection with \(H\), then we say that \(K\) is a \(\oplus\)-complement of \(H\). The minimum possible rank of a \(\vee\)-complement (resp. \(\oplus\)-complement) of \(H\) is called the \(\vee\)-corank (resp. \(\oplus\)-corank) of \(H\). We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the \(\vee\)-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds. -
- Algorithmic recognition of infinite cyclic extensions (with B. Cavallo, D. Kahrobaei, and E. Ventura).
Journal of Pure and Applied Algebra , 221(9), pp. 2157-2179. (2017).We prove that one cannot algorithmically decide whether a finitely presented \(\mathbb{Z}\)-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique \(\mathbb{Z}\)-extensions, and the semi-conjugacy problem for deranged outer automorphisms. -
- Algorithmic problems for free-abelian times free groups (with E. Ventura).
Journal of Algebra , 391(1), pp. 256-283. (2013).We study direct products of free-abelian and free groups with special emphasis on algorithmic problems. After giving natural extensions of standard notions into that family, we find an explicit expression for an arbitrary endomorphism of \(\mathbb{Z}^m \times F_n\). These tools are used to solve several algorithmic problems for \(\mathbb{Z}^m \times F_n\): the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.
Preprints
- The finitely generated intersection property in fundamental groups of graphs of groups (with M. Linton, J. Lopez de Gamiz, M. Roy, and P. Weil).
(2025).A group \(G\) is said to satisfy the finitely generated intersection property (f.g.i.p.) if the intersection of any two finitely generated subgroups of \(G\) is again finitely generated. The aim of this article is to understand when the fundamental group of a graph of groups has the f.g.i.p. Our main results are general criteria for the f.g.i.p. in graphs of groups which depend on properties of the vertex groups, properties of certain double cosets of the edge groups and the structure of the underlying graph. For acylindrical graphs of groups, we also obtain criteria for the strong f.g.i.p. (s.f.g.i.p.). Our results generalise classical results due to Burns and Cohen on the f.g.i.p. for amalgamated free products and HNN extensions. As a concrete application, we show that a graph of locally quasi-convex hyperbolic groups with virtually \(\mathbb{Z}\) edge groups (for instance, a generalised Baumslag--Solitar group) has the f.g.i.p. if and only if it does not contain \(F_2 \times \mathbb{Z}\) as a subgroup. In addition, we show that this condition is decidable. The main tools we use are the explicit constructions of pullbacks of immersions into a graph of groups, obtained by the authors in a previous paper, and a technical condition on coset interactions, introduced in this paper. - Pullbacks and intersections in categories of graphs of groups (with M. Linton, J. Lopez de Gamiz, M. Roy, and P. Weil).
(2025).We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called \(\mathbb{A}\)-product of two morphisms into a graph of groups \(\mathbb{A}\) -- a graph of groups which, within the appropriate categorical setting, captures the intersection of subgroups of the fundamental group of \(\mathbb{A}\). We show that, in the category of pointed graphs of groups, pullbacks always exist and correspond precisely to pointed \(\mathbb{A}\)-products. In contrast, pullbacks do not always exist in the category of unpointed graphs of groups. However, when they do exist, and we show that it is the case, in particular, under certain acylindricity conditions, they are again closely related to \(\mathbb{A}\)-products. We trace, all along, the parallels with Stallings' classical theory of graph immersions and coverings, in relation to the study of the subgroups of free groups. Our results are useful for studying intersections of subgroups of groups that arise as fundamental groups of graphs of groups. As an example, we carry out an explicit computation of a pullback which results in a classification of the Baumslag--Solitar groups with the finitely generated intersection property.
Book chapters
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- Whitehead Problems for Words in \(\mathbb{Z}^m \times F_n\)
Extended Abstracts Fall 2012 , pp. 35-38. (2014).We generically call Whitehead problems for a finitely presented group \(G\) the problems consisting in, given two objects (of a suitable kind) in \(G\) and a family of transformations, deciding whether there exists a transformation sending one object to the other. It is customary to include, in case it exists, the search for such a transformation.
Theses
- Extensions of Free Groups: Algebraic, Geometric, and Algorithmic Aspects
Universitat Politècnica de Catalunya (UPC) , pp. 394. (2017).In this work we use geometric techniques to study natural extensions of free groups and solve several algorithmic problems on them. We consider the family of free-abelian times free groups \(\mathbb{Z}^m \times F_n\) as a seed towards further generalizations in two directions: semidirect products and partially commutative groups. The thesis addresses: (1) structure and algorithmic problems in \(\mathbb{Z}^m \times F_n\) (including subgroup and endomorphism problems); (2) algorithmic recognition and undecidability phenomena for certain cyclic (\(\mathbb{Z}\)-)extensions; (3) extensions of Stallings automata methods to enriched automata for related families; and (4) intersection problems in the setting of Droms groups. - Problemes algorísmics en grups lliures per lliure-abelià
Universitat Politècnica de Catalunya (UPC) , pp. 106. (2011).