Titles and abstracts​

 

Friday, May 17th (room 101)

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• 15:00 - 15:50 Dario Ascari (University of the Basque Country)

The Dehn function of subdirect products of free groups

 

Subgroups of direct products of free groups can be very wild in general; however, they become much more controlled once they are required to satisfy some finiteness condition.

 

We investigate the Dehn function of such groups, which represents the complexity of solving the word problem. We show that, for subgroups of type F_{n-1} in a product of n factors, there is a uniform polynomial bound of N^9 on all the Dehn functions. We also show an example of a subgroup whose Dehn function is exactly N^4; the computation is based on an invariant built using braid groups.

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• 16:00 - 16:50 Dominik Francoeur (Newcastle University)

Intersection-saturated groups and groups with micro-supported actions

 

A group is said to possess the Howson property if the intersection of any two finitely generated subgroups is again finitely generated. When a group fails to have the Howson property, one can try to characterise how far it is from having it. Such considerations give rise to the notion of intersection-saturated groups, introduced recently by Delgado, Roy and Ventura.

 

In this talk, we'll see a new construction for intersection-saturated groups based on so-called micro-supported actions, that allows one in particular to produce finitely presented amenable examples.

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• 17:00 - 17:50 Marco Linton (University of Oxford)

The relation gap and relation lifting problems for groups with cyclic relation modules

 

If F is a free group and F/N is a presentation of the group G, there is a natural way to turn the abelianisation of N into a left ZG-module, known as the relation module of the presentation. A presentation F/N is said to have relation gap if its relation module has strictly fewer ZG-module generators than N has normal generators.

 

Infinite relation gaps were found by Bestvina--Brady, but the existence of finite relation gaps remains an open problem, closely related with questions on the homotopy types of 2-complexes such as Wall's D(2) problem. I will first motivate the problem and survey what is known and what is not known. Then I will present a solution to the relation gap and relation lifting problems for certain groups with cyclic relation module.

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Saturday, May 18th (room 101)

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• 11:00 - 11:50 Yuri Santos (University of Lincoln)

Thompson groups of irrational slope and their Sigma invariants

 

The groups of Richard Thompson – and their relatives or generalisations – became famous by serving as (counter)examples to many interesting questions. Particularly, they tend to have nice cohomological features. In this talk we shall focus on an irrational-slope version of Thompson's group F, introduced by Cleary in the mid 1990s, and some of its cohomological properties. We then discuss the Sigma invariants of Bieri, Neumann, Strebel, and Renz and how to compute them in the case at hand, and potentially state some structural consequences of these computations. 

Based on joint works with Lewis Molyneux and Brita Nucinkis, and with Paula Macedo Lins de Araujo and Altair Santos de Oliveira-Tosti.

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• 12:00 - 12:50 Armando Martino (University of Southampton)

The conjugacy problem for Out(F_3)

 

Dehn's problems have been central to the birth and direction of geometric group theory, and this talk will be mainly concerned with the second of these, the conjugacy problem. This problem asks, for a given group, if there is an algorithm which can determine whether or not two elements of the group are conjugate. I would like to announce a positive solution for a very particular group, Out(F_3), which is the group of outer automorphisms of the free group of rank 3.

 

The problem for general n - that is, for Out(F_n) - remains stubbornly open even though these groups have been the subject of an intense amount of study. I will gently sketch the proof strategy, talk about analogues with the mapping class group of a hyperbolic surface as well GL_n(Z), the group of invertible matrices over the integers, and give an idea of the techniques that we used to solve the problem.

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(Joint work with: F Dahmani, S. Francaviglia and N. Touikan.)