Fall Redbud Topology Conference

The talks are SCEN 408 on the University of Arkansas campus. Abstracts are below.

Time	Speaker	Talk
9:00 − 10:00	Thomas Church	A survey of representation stability
10:15 − 11:15	Matthew Durham	Convex cocompactness and stability in mapping class groups
11:30 − 12:30	Pengcheng Xu	P−moves between pants block decompositions of 3−manifolds
2:30 − 3:30	Matt Day	On the second homology group of the Torelli subgroup of Aut(F_n)
3:45 − 4:45	Robert Tang	Shadows of Teichmueller discs in the curve complex

Abstracts:

Thomas Church, Stanford University
Title: A survey of representation stability
Abstract: I will give a gentle survey of the theory of representation stability, viewed through the lens of its applications in topology and elsewhere. These applications include: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; Charney's theorem and Hecke eigenclasses in stable mod-p cohomology; uniform generators for congruence subgroups and "congruence" subgroups; and distributional stability for random squarefree polynomials over finite fields.

Matt Day, University of Arkansas
Title: On the second homology group of the Torelli subgroup of Aut(F_n)
Abstract: I will discuss joint work with Andrew Putman in which we find an infinite generating set for H₂(IA_n; ℤ), the second homology of the Torelli subgroup (a.k.a. IA_n) of the automorphism group of the free group. This generating set comes from a new infinite presentation for IA_n. There is a natural action of GL(n, ℤ) on H₂(IA_n; ℤ) and our generating set is the orbit of finitely many "basic generators" under this action. Our generators "look the same" for all n starting with n = 6, proving a surjective representation stability conjecture for H₂(IA_n; ℤ).

Matthew Durham, University of Michigan
Title: Convex cocompactness and stability in mapping class groups
Abstract: Originally defined by Farb−Mosher to study hyperbolic extensions of surface subgroups, convex cocompact subgroups of mapping class groups have deep ties to the geometry of Teichmuller space and the curve complex. I will present a strong notion of quasiconvexity in any finitely generated group, called stability, which coincides with convex cocompactness in mapping class groups. I will also discuss some work in progress and future directions. This is joint work with Sam Taylor.

Robert Tang, University of Oklahoma
Title: Shadows of Teichmueller discs in the curve complex
Abstract: Given a flat structure on a surface, we can deform the metric using elements of SL(2,ℝ) to obtain different metrics. The SL(2,ℝ)−orbit under this action is called a Teichmueller disc. The set of systoles arising on some flat metric on a Teichmueller disc gives a quasiconvex subset in the curve graph associated with the surface. We describe nearest point projections to the systole set using a generalised notion of Masur−Minsky's balance time. We will also describe other sets of curves associated to a given Teichmueller disc. This is joint work with Richard Webb.

Pengcheng Xu, Oklahoma State University
Title: P−moves between pants block decompositions of 3−manifolds
Abstract: A pants block decomposition of a compact hyperbolic 3−manifold is a decomposition of the 3−manifold which cuts the manifold into fundamental pieces called pants blocks. This is similar to a triangulation, which cuts the 3−manifold into tetrahedra. In this talk we will discuss how to relate two pants block decompositions of a manifold with a sequences of P−moves, which are similar to Pachner moves between triangulations.