Adam Clay: Generalized torsion and orderability of 3-manifold groups

Abstract: The relationship between left-orderability, bi-orderability, and the existence of generalized torsion in a group is a rather subtle one. While bi-orderable groups are generalized torsion-free, the converse does not hold—but it is surprisingly difficult to construct a group which is generalized torsion-free and not bi-orderable. Motivated by the distinctive behaviour of 3-manifold groups with respect to orderability, after accumulating a significant amount of evidence Motegi and Teragaito conjectured that a 3-manifold group is bi-orderable if and only if it is generalized torsion-free. I'll present a counterexample to this conjecture, and discuss how generalized torsion nonetheless looks to be a promising avenue for understanding left-orderability of 3-manifold groups.

Elizabeth Field: Conformal dimension bounds for certain Coxeter group Bowditch boundaries

Abstract: One of the central problems of geometric group theory is to determine when two finitely-generated groups are quasi-isometric. In this talk, we will explore a family of Coxeter groups with defining graph a complete graph and with each edge label at least 3. Each group in this family is hyperbolic relative to a finite collection of conjugacy classes of abelian subgroups (i.e. is CAT(0) with isolated flats), and by work of Haulmark-Hruska-Sathaye has a Menger curve as its visual boundary. Therefore, these groups cannot be distinguished by the topology of their visual boundary. Nevertheless, we will see that there are infinitely many quasi-isometry classes of groups within this family by finding upper and lower bounds for the conformal dimension of the Bowditch boundary of each group. Along the way, we will see that each of these non-hyperbolic Coxeter groups admits a geometrically finite action on a CAT(-1) model space. This work is joint with Radhika Gupta, Robbie Lyman, and Emily Stark.

Yang Hu: On the enumeration of SU(n)-gauge groups over spheres.

Abstract: Gauge groups are important objects of study in topology. Given a principal G-bundle P over a finite CW complex X, the gauge group Gauge(P) consists of G-equivariant automorphisms of P, and it is a classical question to classify the homotopy types of Gauge(P), as P ranges over the principal G-bundles over X. Of particular interest is the enumeration of the homotopy types of SU(n)-gauge groups over spheres S^m. In this talk, we present lower bounds for this enumeration in the last stable case (when m=2n) and in a family of unstable cases (when m is odd and m>2n). This is joint work with Martin Frankland.

Kim Ruane:  How Topology plays a role in Geometric Group Theory

Abstract:  In Geometric Group Theory, we study infinite groups using geometry. The geometry could be a Cayley graph of the group, or it could be a richer metric space on which the group acts nicely.  In order to study these non-compact geometries, we often compactify them with a boundary to bring problems about the group “to infinity".  In Geometric Group theory, there are many different types of boundaries we can consider in order to see the asymptotic behavior of the group we are interested in. In this way, an algebra question or a geometric question can become a topology question about these boundaries.  A familiar example of this viewpoint is Stallings ends theorem which can be stated loosely as:  a finitely generated group is one-ended if and only if it is connected at infinity. 

We will discuss several different types of boundaries for finitely generated groups and how the topologies of these boundaries uncover information about a group.  

Liam Watson: Mutations, after Dale

Abstract: One of the first things Dale got me hooked on was the Jones polynomial. Related to his quest for a non-trivial knot with Jones polynomial 1, he had me thinking about mutation and other operations that alter knots while leaving the Jones polynomial unchanged. At some point during my MSc, he made the off-hand remark that the Jones polynomial arose as the Euler characteristic of a then newly-discovered homology theory due to Khovanov. This object, and the question of whether it is or is not sensitive to mutation, occupied me off and on ever since. It turns out that Khovanov’s reduced invariant is, in fact, insensitive to mutation. I’ll explain some of this story, which is part of joint work with Artem Kotelskiy and Claudius Zibrowius, and single out a surprising application of homological mirror symmetry for the 3-punctured sphere.

Shangjie Zhang: RO(C₂)-graded Hurewicz images and C₂ equivariant Adams differentials

Abstract: We completely determine the RO(C₂)-graded Hurewicz images of H𝔽̲₂̲, Hℤ̲, kℝ, and ko_C₂. Surprisingly, the answer is closely tied to the problem of vector fields on spheres. Moreover, at any filtration in the genuine C₂-equivariant Adams spectral sequence and at any nonzero filtration of the ℝ-motivic Adams spectral sequence, we show that there exist classes supporting nonzero differentials of arbitrary length. This talk contains joint work with Guo, Kong, Li, Lu, Ma, Shi, Wu, Xu, and Yang.

We gratefully acknowledge the support of the Department of Mathematics of the University of British Columbia,  the National Sciences and Engineering Research Council of Canada,   the Pacific Institute for Mathematical Science, and the U.S. National Science Foundation.