Fall Redbud Topology Conference

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

Time	Speaker	Talk
9:00 − 9:50	Jeffrey Meyer	Geodesics & surfaces: a rigid interaction
10:05 − 10:55	Corey Bregman	Automorphisms of cube complexes and the Torelli subgroup of a right-angled Artin group
11:10 − 12:00	Chaim Goodman-Strauss	Aperiodic tiling
2:00 − 2:50	Andy Eisenberg	Recognizing right-angled Coxeter groups
3:05 − 3:55	Susan Hermiller	Trees, flow functions, and algorithms for groups
4:10 − 5:00	Justin Malestein	Arithmetic quotients of the mapping class group from covers

Abstracts:

Corey Bregman, Rice University
Title: Automorphisms of cube complexes and the Torelli subgroup of a right-angled Artin group
Abstract: Let G be a right-angled Artin group (raag) and let Out(G) be its outer automorphism group. The Torelli subgroup of Out(G) is the kernel of the action of Out(G) on the abelianization of G. Recently, Charney−Stambaugh−Vogtmann constructed an outer space for a subgroup of Out(G), whose vertices correspond to certain cube complexes called blow-ups of Salvetti complexes. Using this outer space, we present a geometric proof that the Torelli subgroup of Out(G) is torsion-free. The proof relies on understanding the combinatorial geometry of blow-ups of Salvetti complexes.

Andy Eisenberg, Oklahoma State University
Title: Recognizing right-angled Coxeter groups
Abstract: Right-angled Coxeter groups are a class of groups that have been widely studied because of their nice combinatorial, algorithmic, and geometric properties. However, there are not many methods for detecting right-angled Coxeter-ness of a group, and even fewer for constructively finding right-angled Coxeter presentations of the group. I will discuss a new method and some first applications, developed in joint work with Charlie Cunningham, Adam Piggott, and Kim Ruane.

Chaim Goodman-Strauss, University of Arkansas
Title: Aperiodic tiling
Abstract: Aperiodic sets of tiles, most famously the Penrose tiles, disrupt global group structure through the use of local matching rules—for example, there are uncountably many combinatorially distinct Penrose tilings, yet none are periodic. In recent years, there has been growing activity, with constructions in a wider variety of settings, such as the Heisenberg group, or hyperbolic n−space. We survey many recent results and touch on several open problems.

Susan Hermiller, University of Nebraska
Title: Trees, flow functions, and algorithms for groups
Abstract: A finitely presented group has a bounded flow function (i.e. is stackable) if there is a maximal tree in the Cayley graph and a map from edges outside of the tree to directed paths of bounded length, such that the iteration of this map on any edge results in a path lying completely within the tree after finitely many steps. When the graph of this flow function is recognizable by a finite state automaton, the resulting autostackable group has solvable word problem, but not necessarily solvable conjugacy problem. In this talk I'll discuss many examples of stackable and autostackable groups. This includes joint work with Mark Brittenham, Derek Holt, Ashley Johnson, and Conchita Martinez.

Justin Malestein, University of Oklahoma
Title: Arithmetic quotients of the mapping class group from covers
Abstract: In this talk, I will discuss a procedure for obtaining infinitely many "virtual" arithmetic quotients of mapping class groups of closed surfaces, (surjective maps up to finite index). The representations arise naturally from lifting elements of the mapping class group to covers of the original surface where they act on the homology. By choosing covers appropriately, one can produce arithmetic quotients of type Sp(2m), SO(2m, 2m), and SU(m, m) for arbitrarily large m in every genus. Joint with F. Grunewald, M. Larsen, and A. Lubotzky.

Jeffrey Meyer, University of Oklahoma
Title: Geodesics & surfaces: a rigid interaction
Abstract: In this talk, I will discuss recent joint work with Benjamin Linowitz and Paul Pollack on the interplay between closed geodesics and totally geodesic surfaces in arithmetic hyperbolic 3−manifolds. It has been shown that the commensurability class of an arithmetic hyperbolic 3−manifold is completely determined by the set of lengths of closed geodesics (called the length set) as well as by the set of totally geodesic surfaces. It is natural to ask whether the commensurability class is determined by only the lengths of closed geodesics lying on totally geodesic surfaces. We answer this question and go on to quantify (using counting techniques from analytic number theory) the difference between this "totally geodesic length set" and the length set. To explain these results, I will give a gentle introduction to the theory of arithmetic hyperbolic 3−manifolds and explain how manifolds, their surfaces, and their geodesics relate to division algebras, their subalgebras, and their maximal subfields.