Elements of a finitely generated group have a natural notion of length: namely the length of a shortest word over the generators that represents the element. This allows us to study the growth of such groups by considering the size of spheres with increasing radii. One current area of interest is the rationality or otherwise of the formal power series whose coefficients are the sphere sizes. I will describe a combinatorial way to study this series for the class of virtually abelian groups, introduced by Benson in the 1980s, and then outline its applications to other types of growth series.

# Past Junior Topology and Group Theory Seminar

Thompson's group F is a group of homeomorphisms of the unit interval which exhibits a strange mix of properties; on the one hand it has some self-similarity type properties one might expect of a really big group, but on the other hand it is finitely presented. I will give a proof of finite generation by expressing elements as pairs of binary trees.

When dealing with geometric structures one natural question that arise is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometric invariant. This motivates the notion of Morse subsets. In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups (HHG), and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell.

In 2012 Eskin, Fisher and Whyte proved there was a locally finite vertex transitive graph which was not quasi-isometric to any connected locally finite Cayley Graph. This motivates the study of vertex transitive graphs from a geometric group theory point of view. We will discus how concepts and problems from group theory generalise to this setting. Constructing one framework in which problems can be framed so that techniques from group theory can be applied. This is work in progress with Agelos Georgakopoulos.

TBC

The $n$-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus $g$ with $n$-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the $n=1$ case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of $g$ and $n$.

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.

Cube Complexes with Coupled Links (CLCC) are a special family of non-positively curved cube complexes that are constructed by specifying what the links of their vertices should be. In this talk I will introduce the construction of CLCCs and try to motivate it by explaining how one can use information about the local geometry of a cube complex to deduce global properties of its fundamental group (e.g. hyperbolicity and cohomological dimension). On the way, I will also explain what fly maps are and how to use them to deduce that a CAT(0) cube complex is hyperbolic.