Here you can find the list of speakers of the lightning talks. All talks will take place in Room 201 of the Physical Sciences Center
(PHSC), located at 601 Elm Avenue.
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Title: Free Factor Homology of Finite Covers of Graphs
Abstract: In a 2016 paper, Benson Farb and Sebastian Hensel ask if the homology of a finite cover of a graph \(\Gamma\) is generated by lifts of primitive elements of \(\pi_1(\Gamma)\). Surprisingly, the answer is no: in fact, by a theorem of Malestein and Putman from 2019, if the fundamental group of \(\Gamma\) is nonabelian, then there is a finite cover \( \tilde{\Gamma}\) of \(\Gamma\) whose primitive homology is infinite index inside \(H_1(\tilde{\Gamma})\).
We are motivated by a question of Autumn Kent, which asks, given a possibly punctured surface \(\Sigma\) with fundamental group of rank at least three, if the homology of every finite cover of \(\Sigma\) is generated by components of lifts of nonfilling curves in \(\Sigma\). If the answer to Kent's question is yes, then the homology of any finite cover of a graph \(\Gamma\) is generated by components of lifts of curves in a proper free factor of \(\pi_1(\Gamma)\). For any finite cover \(\tilde{\Gamma}\) of \(\Gamma\), we construct a space whose connectedness would imply \(H_1^{\text{pff}}(\tilde{\Gamma}) = H_1(\tilde{\Gamma})\), where \(H_1^{\text{pff}}(\tilde{\Gamma})\) is the subspace of \(H_1(\tilde{\Gamma})\) generated by components of lifts of curves in a proper free factor of \(\pi_1(\Gamma)\).
Title: Large Scale Geometry of Pure Big Mapping Class Groups
Abstract: Motivated by the tools of Mann and Rafi, we show the pure mapping class groups of infinite type surfaces with more than one end are not CB generated and thus cannot be given a well defined quasi-isometry type in this way. In particular, we prove that PMap(S) is globally CB, locally CB, or CB generated, if and only if S is the Loch Ness monster surface.
Title: Skein Theory of Affine Type A Subfactor Planar Algebras
Abstract: Subfactor planar algebras first appeared out of studying the standard invariant of a subfactor. Planar algebras can be conveniently encoded by diagrams in the plane. These diagrams satisfy some skein relations and have an invariant called an index. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and give presentations for affine type A subfactor planar algebras of index 4.
Title: Automorphisms of the fine 1-curve graph
Abstract: The fine curve graph of a surface S was introduced by Bowden–Hensel–Webb in 2019 to study the diffeomorphism group of S. We build on the works of Long–Margalit–Pham–Verberne–Yao and Le Roux–Wolff to show that the automorphism group of the fine 1-curve graph, a variant of the fine curve graph, is the homeomorphism group of S. This work is joint with K. W. Booth and D. Minahan.
Title: Property R∞ for Artin groups
Abstract: A group G has property R∞ if for every automorphism φ of G the number of twisted φ-conjugacy classes is infinite. This property is motivated by the topological fixed point theory, and has been a subject of active research. Among the groups which have this property are hyperbolic and relatively hyperbolic groups, mapping class groups, generalized Baumslag-Solitar groups and some others. However, the general picture of which groups have this property is quite elusive. In a joint project with Matthieu Calvez, we establish property R∞ for some spherical and affine Artin groups by utilizing their close relation to certain mapping class groups of punctured surfaces.
Title: Complexes of Groups and the Developability Criterion
Abstract: It is a known fact that a graph of groups has an associated space known as the Bass-Serre tree on which the fundamental group G acts on without inversion. The graph of groups corresponding to this action is the graph of groups we started out with. For the case of complexes of groups, this is not always true. In this talk, I will introduce the notion of a complex of groups and touch on the developability criterion, a necessary and sufficient condition for a complex of groups to be associated to an action of a group on a polyhedral complex.
Title: Short Closed Geodesics with Self-Intersections
Abstract: We consider the set of closed geodesics on a hyperbolic surface. Given any non-negative integer k, we are interested in the set of closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.
Title: The Isomorphism Problem for Small Rose Generalized Baumslag-Solitar Groups
Abstract: Generalized Baumslag-Solitar groups are finitely generated groups that act on trees with infinite-cyclic vertex and edge stabilizers. The n-rose GBS groups are GBS groups such that the underlying quotient graph of groups look like a rose with n petals. Recently these groups have been studied in many aspects such as boundary actions and C*-simplicity. In this talk, we will talk about the isomorphism problem for n-rose GBS groups.