Past Junior Topology and Group Theory Seminar

13 June 2018
16:00
Nicolaus Heuer
Abstract

We will study the l1-homology of the 2-class in one relator groups. We will see that there are many qualitative and quantitive similarities between the l1-norm of the top dimensional class and the stable commutator length of the defining relation. As an application we construct manifolds with small simplicial volume.

This work in progress is joint with Clara Loeh.

  • Junior Topology and Group Theory Seminar
6 June 2018
16:00
Alex Margolis
Abstract

One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse PD_n subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This  generalises work of Whyte and Mosher--Sageev--Whyte.

  • Junior Topology and Group Theory Seminar
30 May 2018
16:00
Esmee te Winkel
Abstract

In the 80s, Hatcher and Thurston introduced the pants graph as a tool to prove that the mapping class group of a closed, orientable surface is finitely presented. The pants graph remains relevant for the study of the mapping class group, sitting between the marking graph and the curve graph. More precisely, there is a sequence of natural coarse lipschitz maps taking the marking graph via the pants graph to the curve graph.

A second motivation for studying the pants graph comes from Teichmüller theory. Brock showed that the pants graph can be interpreted as a combinatorial model for Teichmüller space with the Weil-Petersson metric.

In this talk I will introduce the pants graph, discuss some of its properties and state a few open questions.

  • Junior Topology and Group Theory Seminar
23 May 2018
16:00
Abstract

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.

  • Junior Topology and Group Theory Seminar
16 May 2018
16:00
Sam Shepherd
Abstract

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.

  • Junior Topology and Group Theory Seminar
10 May 2018
16:00
Davide Spriano
Abstract

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.

  • Junior Topology and Group Theory Seminar
2 May 2018
16:00
Alexander Wendland
Abstract

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.

  • Junior Topology and Group Theory Seminar
7 March 2018
16:00
Marissa Loving
Abstract

 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 $n$.

  • Junior Topology and Group Theory Seminar

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