Titles & Abstracts

The conjugacy problem for a group corresponds to solving equations of the form gX=Xh. In this talk, I will explain how to solve the conjugacy problem for ascending HNN-extensions of free groups. In 2006, Bogopolski+Martino+Maslakov+Ventura solved the conjugacy problem for free-by-cyclic groups. Their proof is based on 2 key components, which are both proven using an analysis of free groups automorphisms via train-track maps. We follow this same route, but instead use an analysis of free group endomorphisms via the "automorphic expansions'' of Mutanguha to prove the analogous 2 key components.

In this talk I will present a family of 23 torsion free, non-right-angled hyperbolic groups that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness, and discuss the search for surface subgroups in these groups. This research is naturally motivated by Gromov’s famous surface subgroup question: Does every one-ended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2? For non-right-angled groups the answer to this question is still unknown. Together with Alina Vdovina, Jarkko Savela, Emilia Oikarinen, and Matti Järvisalo we have shown that only seven of these 23 hyperbolic groups have periodic apartments of genera 2, 3, or 4, and thus posses surface subgroups of genus less than five.

This talk will be a discussion of infinite groups which share properties with finite groups, either in their actions (strongly bounded groups) or in their relationship to proper subgroups (Jonsson groups). I will review some history and present some recent constructions of such groups. Includes joint work with Saharon Shelah.

“What can one describe by first-order formulas in a given structure A?” - is an old and interesting question. Of course, this depends on the structure A. For example, in a free group only cyclic subgroups (and the group itself) are definable in the first-order logic, but in a free monoid of finite rank any finitely generated submonoid is definable. A structure A is called rich if the first-order logic in A is equivalent to the weak second order logic. Surprisingly, there are a lot of interesting groups, rings, semigroups, etc., which are rich. I will talk about some of them and then describe various algebraic, geometric, and algorithmic properties that are first-order definable in rich structures. Weak second order logic can be introduced into algebraic structures in different ways: via HF-logic, or list superstructures over A, or computably enumerable infinite disjunctions and conjunctions, or via finite binary predicates, etc. I will describe a particular form of this logic which is especially convenient to use in algebra and show how to effectively translate such weak second order formulas into the equivalent first-order ones in the case of a rich structure A.

The theory of real trees and groups acting on them has had a deep impact on Group Theory by providing tools to attack new problems, by simplifying proofs of classical results, and by establishing new connections between group theory and geometry, topology, dynamical systems and model theory.

In this talk, we will introduce a new class of metric spaces, called real cubings, which we view as higher-dimensional real trees. We will describe their structure and characterise them from different viewpoints.

As hyperbolic groups are linked to real trees via their asymptotic cone, we will show that real cubings are connected to hierarchically hyperbolic groups, a class of groups that contains right-angled Artin groups and the mapping class groups of closed surfaces.

We will then speculate why we believe that a good theory of groups acting on real cubings is possible. The talk is based on joint work with Montserrat Casals-Ruiz and Mark Hagen.

We describe the structure of non full-sized normal subgroups in the automorphism group of a right-angled Artin group $Aut(A_\Gamma)$. In particular, we prove that a finite normal subgroup in $Aut(A_\Gamma)$ has at most order two and if $\Gamma$ is not a clique, then any finite normal subgroup in $Aut(A_\Gamma)$ is trivial. This property has an implication to automatic continuity: every algebraic epimorphism $\varphi\colon L\twoheadrightarrow{\rm Aut}(A_\Gamma)$ from a locally compact Hausdorff group $L$ is continuous if and only if $\Gamma$ is not a clique. For ${\rm Aut}(G_\Gamma)$ where $G_\Gamma$ is a graph product of (cyclic) groups, we obtain similar results.

In recent years, the discrete Bakry-Émery theory on graphs has become an active emerging research field. Fan Chung and S. T. Yau showed that all abelian Cayley graphs are non-negatively curved. However apart from some special families, such as Coxeter groups, the curvature properties of non-abelian Cayley graphs are not known. In this talk we reformulate the Bakry-Émery curvature on a graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. We view this curvature matrix as the discrete Ricci curvature tensor. Using this reformulation we will calculate the curvature of various classes of Cayley graphs including hybrids of Coxeter and Artin groups. We end by showing that adding a relation to a Cayley graph does not decrease curvature.

Just before the pandemic, I wrote an article for the Baumslag memorial volume in which I presented some novel constructions concerning the homology and profinite completions of finitely generated groups. I shall describe some of those results here. In particular, I shall explain why there exists a finitely presented acyclic group U such that U has no proper subgroups of finite index and every finitely presented group can be embedded in U. I shall also explain why there is no algorithm that can determine whether or not a finitely presentable subgroup of a residually finite, biautomatic group is perfect. And if time allows, I shall explain a method for generating torsion in the profinite completions of torsion-free groups.

A typical (and usually hard) problem in a group $G$ consists on deciding whether a given equation has solutions and, in the affirmative case, find them all. For the free group, this is a famous problem solved by Makanin and Razborov several decades ago. In this talk, again in a free group $F$, we solve the dual problem: given an element $g\in F$ and a finitely generated subgroup $H<F$, we decide whether or not $g$ satisfies some non-trivial equation over $H$ and, in the affirmative case, we'll find them all (we'll prove they form a normal subgroup finitely generated as such, and will compute such a set of finite normal generators). We give two alternative algorithms: one based on classical techniques (Nielsen transformations, etc) and another one based in Stallings graphs. (Joint work with A.,Rosenmann).

In this talk, we will discuss subgroups of right-angled Artin groups (RAAGs for short). Although in general, subgroups of RAAGs are known to have a wild structure and bad algorithmic behaviour, in this talk, we will center on properties of subgroups and specific families of RAAGs that assure a tame structure and good algorithmic behaviour.

More precisely, we will discuss finitely generated normal subgroups. A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. We will discuss a generalisation of this result and show that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As a corollary, we deduce, among others, that finitely generated normal subgroups of RAAGs have decidable word, conjugacy and membership problems and that they are hereditarily conjugacy separable.

Secondly, we will recall results of Baumslag-Roseblade and Bridson-Howie-Miller-Short on subgroups of the direct product of free groups and explain how they generalise to other classes of RAAGs. We will show that in these classes, finitely presented subgroups have a tame structure and that the algorithmic problems are also decidable.

This is joint work with Jone Lopez de Gamiz Zearra.

In 2016, Ciobanu, Diekert and Elder showed that sets of solutions to systems of equations in free groups can be expressed as EDT0L languages, giving a bound on the space complexity. Since then, these ideas have been extended to various classes of groups including hyperbolic groups, right-angled Artin groups and virtually abelian groups. I will discuss recent work on expressing solutions to single equations in the Heisenberg group as EDT0L languages, and the relationship this problem has to quadratic Diophantine equations in the ring of integers and Pell’s equation.

The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this talk, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups.

For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.

This is joint work with A.Yu.Olshanskii

After Kharlampovich-Myasnikov and Sela proved that nonabelian free groups share the same first-order theory, the model theoretic interest for the subject arose. A historically important question for any natural first-order theory is whether it interprets an infinite field. In this talk I will explain some of the principal ideas in proving that no infinite field is interpretable in the first-order theory of nonabelian free groups.

In this talk I will discuss group equations with abelian predicates, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications.

I will introduce equations in groups and length constraints, and then focus on abelianisation constraints (or predicates). I will show that the problem of solving equations with abelian predicates is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation. By contrast, in groups with finite abelianisation where equations are decidable, adding abelian constraints to the problem remains decidable. This is joint work with Albert Garreta.