Fall 2017 Redbud Topology Conference

The talks are in Price Hall room 2040 on the University of Oklahoma campus. Abstracts are below.

Time	Speaker	Talk
8:30 − 9:00	coffee + bagels
9:00 − 10:00	Robert Haraway	Tessellations of two higher Teichmuller spaces
10:15 − 11:15	Yo'av Rieck	Admitting an S³ Dehn filling is NP-hard
11:30 − 12:30	Justin Malestein	Finite covers of surfaces and simple closed curves
	lunch
2:30 − 3:30	Ignat Soroko	Stable commutator length in two-dimensional right-angled Artin groups
3:45 − 4:45	Amey Kaloti	Hyperbolic 3-manifolds not admitting fillable contact structures

Abstracts:

Robert Haraway, Oklahoma State University
Title: Tessellations of two higher Teichmuller spaces
Abstract: Cooper and Long introduced an analog of Penner's decomposition of Teichmuller space for singly-cusped surfaces, and asked if it was a cell decomposition. Their construction generalizes easily to a decomposition of Fock and Goncharov's space A⁺ for SL₃ over R for multiply cusped surfaces. In joint work with Stephan Tillmann, we have shown that Cooper and Long's decomposition is a cell decomposition for the A-space of the once-punctured torus and for the thrice-punctured sphere. We have also shown that the Penner-Weeks procedure for determining isometry of two decorated hyperbolic surfaces admits an analog in this setting as well.

We give an introduction to these ideas and results, motivating them by analogy with hyperbolic geometry, and sketch the main ideas in the proof of our result. We conclude with some open questions.

Amey Kaloti, University of Arkansas
Title: Hyperbolic 3-manifolds not admitting fillable contact structures
Abstract: We exhibit an infinite family of hyperbolic rational homology spheres not admitting any fillable contact structure. Relation to existence of tight contact structure on hyperbolic manifolds will be discussed. Joint work with Bulent Tosun.

Justin Malestein, University of Oklahoma
Title: Finite covers of surfaces and simple closed curves
Abstract: In this talk, I will discuss examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. Additionally, I will discuss analogous results for primitive elements and the homology of finite index subgroups of a free group. I will also present a couple consequences of these results including a theorem that Out(F_n) modulo the group generated by kth powers of transvections often has infinite order elements. This is joint work with Andrew Putman.

Yo'av Rieck, University of Arkansas
Title: Admitting an S³ Dehn filling is NP-hard
Abstract: We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into R³ is NP-hard. By a happy coincidence the complex we construct is a 3-manifold with boundary tori, showing also that the problem of deciding whether a 3-manifold with boundary tori admits an S³ Dehn filling is NP-hard.

This is joint work with Arnaud de Mesmay, Eric Sedgwick, and Martin Tancer.

Ignat Soroko, University of Oklahoma
Title: Stable commutator length in two-dimensional right-angled Artin groups
Abstract: The stable commutator length (scl) of an element in a group is a remarkable numerical invariant, which has relevance in several areas of low-dimensional topology, bounded cohomology and dynamics. In general, scl is very hard to compute, but for many important classes of groups it has been shown that the spectrum of possible values of scl has a gap above zero. In particular, Culler showed that for an arbitrary element g of a free group, scl(g) is at least 1/6. In this joint work with Max Forester and Jing Tao, we adapt Culler's approach to the case of right-angled Artin groups whose defining graphs do not have triangles. As a result, we get for arbitrary elements g of such RAAGs the estimate: scl(g) ≥ 1/20.