North British Geometric Group Theory Seminar

Wednesday 14th February 2024, University of Newcastle

• Caterina Campagnolo (Univesidad Autonoma de Madrid), A new vanishing criterion for bounded cohomology

  Bounded cohomology is a powerful and by now well-established tool to study properties of groups and spaces. However, it is often difficult to compute and, beyond the case of amenable groups, many questions remain widely open. In joint work with Francesco Fournier-Facio, Yash Lodha and Marco Moraschini, we present a new algebraic condition that implies the vanishing of the bounded cohomology of a given group for a big family of coefficient modules. This condition is satisfied by many non-amenable groups of topological, geometric or algebraic origin.

• Maria Cumplido Cabello (Universidad de Sevilla), A quadratic solution to the word problem in Artin groups without braid relations

  Artin groups are the groups defined by a finite set of generators and relations of the form sts...=tst... where s and t are generators and both words of the equality have the same length. Despite these groups are easily defined, they are quite mysterious: basic problems of classic group theory remain open, as it is the case for the word problem. There have been many (geometric and algebraic) approaches to solve the word problem for particular families of Artin groups (being the braid group the flagship example). In this talk we will explain a method of rewriting words that allows us to obtain geodesic representatives for elements in Artin groups that do not have a relation of length 3 (also known as braid relations) and, as a direct consequence, we will solve the word problem in this (big) family of Artin groups in quadratic time. This is a joint work with Rubén Blasco-García and Rose Morris-Wright.

• Murray Elder (University of Technology, Sydney) How hard is it to know if there is an epimorphism from one group to another?

  Let $\mathcal C, \mathcal D$ be classes of finitely presented groups. The \emph{epimorphism problem} denoted $\operatorname{Epi}(\mathcal C,\mathcal D)$ is the following decision problem. %\compproblem[]{Group presentations \mee{(or sometimes mult table, etc)} $G \in \cC$ and $H \in \cD$}{Is there an epimorphism $G \twoheadrightarrow H$?} \compproblem[]{Finite descriptions (presentation, multiplication table, other) for groups $G \in \mathcal C$ and $H \in \mathcal D$}{Is there an epimorphism $G \twoheadrightarrow H$?} I will discuss some cases where it is decidable and where it is $\mathsf{NP}$-complete. Spoiler alert: it is undecidable for $C=D=$ the class of 2-step nilpotent groups (Remeslennikov). This is joint work with Jerry Shen (UTS) and Armin Wei\ss (Stuttgart).

• 1200 onwards. Informal meeting for lunch, at Courtyard restaurant on the University of Newcastle campus (That's in the Old Library Building, and the entrance is opposite Merz Court. If you walk onto campus from haymarket metro, you need to walk through the archway and into the main quadrangle, then out of it on the far righthand side; the door you want is immediately on your right).
• 1330-1430 Caterina Campagnolo
• 1445-1545 Maria Cumplido
• 1545-1615 Tea
• 1615-1715 Murray Elder
• 1745 onwards. Early dinner at Ottoman restaurant in Clayton street, close to station

  Lectures will be in room TR1 on the fourth floor of the Herschel buildings (home of the School of Mathematics and Statistics, next to Haymarket) at Newcastle University.

  Please let us know if you plan to stay for dinner, since the restaurant needs an idea of numbers; bearing in mind that 14th Feb is Valentine's day, we have already booked for 20, but need to be try and be accurate.

  The meeting is funded by grants from the London Mathematical Society and the Glasgow Mathematical Journal Learning and Research Support Fund.

  

  Local organiser: Andrew Duncan (AndrewDOTDuncanATnewcastleDOTacDOTuk) and Sarah Rees (SarahDOTReesATnewcastleDOTacDOTuk)