Subsidiary society of:

Geometric and Algorithmic Aspects of Groups

ORG: Andre Carvalho, University of Évora – Jordi Delgado, UPC – Mallika Roy, Harish-Chandra Research Institute

GENERAL PRESENTATION 

This session focuses on the interplay between Geometric and Algorithmic Group Theory. It aims to explore recent developments concerning the relationship between the geometry of groups and their algorithmic behavior, with particular emphasis on the solvability and complexity of decision problems. With its classical origins in the Word, Conjugacy, and Isomorphism Problems, Algorithmic Group Theory now addresses a broad range of computational questions about groups. Frequently studied topics include equations over groups, subgroups, homomorphisms, and dynamical phenomena. The scope of the session includes both theoretical developments and computational methods, ranging from the study of geometric constructions and combinatorial techniques to the development of practical algorithms for investigating infinite groups. This combination of geometric, combinatorial, and computational approaches often provides deeper insight into the structure of groups and yields purely algebraic results. Connections with related areas of geometry, group dynamics, formal language theory, and combinatorics will also be explored. By bringing together these perspectives, the session aims to reflect the vitality of current research in the field and to provide a platform for presenting new results on the geometric and algorithmic aspects of discrete groups.

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SCHEDULE

Dimecres 7 d’octubre — 16:30–18:30 (Pi i Sunyer)

Finite quotients of spherical Artin groups, by Kaitlin Ragosta (University of the Basque Country)

Proving non-isomorphism of two quotients of an Artin group. by Sarah Rees (University of Newcastle)

Membership problems in metabelian groups, by Ruiwen Dong (University of Oxford)

 

Dijous 8 d’octubre — 16:00–18:00 (Nicolau d’Olwer)

Asymptotic Assouad-Nagata dimension of graphical C(3)-T(6) small cancellation complexes, by Damian Osajda (University of Wroclaw)

On the Dehn function of Thompson’s group V, by Dario Ascari (University of the Basque Country)

Finitely generated subgroups of plain groups, by Maria Ana Barbosa (University of Porto)

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ABSTRACTS

Speaker: Dario Ascari.
 
Title: On the Dehn function of Thompson’s group V.
 
Abstract: The Dehn function of a finitely presented group is an invariant based on counting how many relations are needed (at least) to pass from a word to another one representing the same element of the group. This is naturally related to the word problem for the group. It’s a classical result that Thompson’s groups F has quadratic Dehn function, and recent developments show that Thompson’s group T also has quadratic Dehn function; however, the Dehn function of V is still unknown. We provide an upper bound of n^2(log n)^2 on the Dehn function of V, improving substantially the previous results in literature.
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Speaker: Kaitlin Ragosta Hughes.
 
Title: Finite quotients of spherical Artin groups.
 
Abstract: In 2023, Kolay showed that the smallest non-abelian quotient of the braid group on n strands is the symmetric group Sym(n), resolving a long-standing question of Margalit. This naturally raises the questions of whether the smallest non-abelian quotient of an irreducible spherical Artin group is the smallest non-abelian quotient of the corresponding Coxeter group and whether irreducible spherical Artin groups can distinguished by their finite quotient groups. In this talk, I will present work which answers both questions affirmatively. This talk is based on joint work with Sam Hughes, Thomas Ng, Nancy Scherich, and Yvon Verberne.
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Speaker: Maria Ana Barbosa.
 
Title: Finitely generated subgroups of plain groups.
 
Abstract: Plain groups, that is, finite free products of finite groups and
copies of Z, and their finitely generated subgroups have received increasing
attention in recent years. Motivated by the success of Stallings automata in
studying finitely generated subgroups of free groups, we explore their extension
to the class of plain groups. Using this approach, we show that several decision
problems in this class of groups are decidable, and that many properties of a
finitely generated subgroup of a plain group can be inferred directly from its
associated automaton. In this talk, we will present the main ideas behind this
construction and discuss some of the results obtained from it.
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Speaker: Ruiwen Dong.
 
Title: Membership problems in metabelian groups
 
Abstract: In this talk we survey the decidability landscape of membership problems in metabelian groups, in particular Subgroup Membership, Submonoid Membership, and Rational Subset Membership. We will talk about several decidability and undecidability results, and briefly discuss the main ideas and connections between them. The tools leading to these results are highly diverse: they range from commutative algebra and number theory, to combinatorics and automata theory. In the end, we will mention several open problems in this area.

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Speaker: Sarah Rees

Title: Proving non-isomorphism of two quotients of an Artin group

Abstract: I’ll discuss two different proofs
that, for $n \geq 5$, \[ A(D_n) \not\cong G([1,w_1]), \]  where $A(D_n)$ is the Artin group of type $D_n$,  $w_1$ is a proper quasi-Coxeter elt in $W(D_n)$, and $G([1,w_1])$ is
the associated interval group. The question arose from an investigation with Baumeister and Neaime that was looking for a generalisation of the dual approach to Artin groups.

Each group can be found as a quotient of an $n$-generator Artin group (the same one, which we call $A(\Delta_{r,s})$) by a single commutator relation.
Ultimately we hope to identify more properties of the second group from that
presentation.

The first proof of non-isomorphism, of myself with Baumeister, Holt, and Neaime (2023), uses standard methods from computational group theory. Recoognition of patterns in the infinite sequence of proofs for $n\geq 5$ allows combination into a single proof for all those $n$. But the calculation is heavy. And Luis Paris suggested to us that we could prove the result much more easily by exploiting the fact that the braid group $\cB_n$ is a subgroup of the Artin group $A(\Delta_{r,s})$ and then using properties of that Braid group.

The second proof, of myself with Baumeister and Paris (2026), is straightforward,  given some knowledge of properties of the braid group $\cB_n$ on $n$ strands, considered as the mapping class group of the $n$-punctured disc.

I’ll define all the objects under consideration and explain why we were interested to investigate these isomorphisms. Then I’ll explain the basic structure of the two proofs.

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Speaker: Damian Osajda.

 
Title: Asymptotic Assouad-Nagata dimension of graphical C(3)-T(6) small cancellation complexes

Abstract: We establish a universal upper bound on the asymptotic Assouad–Nagata dimension of simply connected, uniformly locally finite graphical C(3)-T(6) small cancellation complexes. More generally, we prove such a bound for systolic complexes with flat intervals, from which the small cancellation case follows.
Joint work with Martín Blufstein, Victor Chepoi, and Huaitao Gui.