Abstracts

Yael Algom-Kfir - The boundary of a hyperbolic free-by-cyclic group
Abstract: Given an automorphism \phi of the free group F_n consider the HNN extension G_φ = F_n \rtimes Z. We compare two cases:

1. φ is induced by a pseudo-Anosov map on a surface with boundary and of non-positive Euler characteristic. In this case G is a CAT(0) group with isolated flats and its (unique by Hruska) CAT(0)-boundary is a Sierpinski Carpet (Ruane).
2. φ is atoroidal and fully irreducible. Then by a theorem of Brinkmann G_φ is hyperbolic. If φ is irreducible then its boundary is homeomorphic to the Menger curve (M. Kapovich and Kleiner).

We prove that if φ is atoroidal then its boundary contains a non-planar set. Our proof highlights the differences between the two cases above. This is joint work with A. Hilion and E. Stark.

Matt Clay - Right-angled Artin groups as normal subgroups of mapping class groups
Abstract: I will talk a method to construct normal subgroups isomorphic to non-free right-angled Artin groups. This construction recovers, expands, and makes constructive the result of Dahmani, Guirardel, and Osin about free normal subgroups generated by a high power of a pseudo-Anosov. We do this by creating a version of their “windmill” construction tailor-made for the projection complexes introduced by Bestvina, Bromberg, and Fujiwara. This is joint work with Johanna Mangahas and Dan Margalit.

Spencer Dowdall - The Scott-Swarup property in hyperbolic group extensions
Abstract: In 1990 Peter Scott and Gadde Swarup proved that, in the fundamental group of a closed, hyperbolic fibered 3-manifold, every finitely generated infinite-index subgroup of the fiber group is quasi-convex in the ambient group. Since then, various work of Kent, Leiniger, Mj, Rafi, Taylor, and myself has shown that this phonomenon in fact holds in many hyperbolic group extensions. This talk will explore this "Scott-Swarup property" in the particular context of free group extensions. We will see how this is related to the fiber subgroup having primitive elements that are qi-embedded with uniformly boudned width, and how both of these follow from nuanced flaring properties in the group. Joint work with Sam Taylor.

Michael Handel - The conjugacy problem for UPG outer automorphisms of F_n
Abstract: This is joint work with Mark Feighn. I will discuss our solution to the conjugacy problem for rotationless polynomially growing outer automorphisms of F_n.

Camille Horbez - Abstract commensurators of some normal subgroups of Out(Fn)
Abstract : We give a new proof of a theorem of Farb and Handel stating that for all n\ge 4, Out(Fn) is equal to its own abstract commensurator. In other words, every isomorphism between finite-index subgroups of Out(Fn) is the restriction of the conjugation by some element of Out(Fn). Our proof enables us to extend their theorem to the n=3 case. More generally, we prove that many normal subgroups of Out(Fn), for example the Torelli subgroup IA_n and all subgroups in the Johnson filtration of Out(Fn), have Out(Fn) as their abstract commensurator. This is a joint work with Martin Bridson and Ric Wade.

Christopher Leininger - Polygonal billiards, Liouville currents, and rigidity
Abstract: A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, in which we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon. This is consequence of our main result about Liouville currents on surfaces associated to nonpositively curved Euclidean cone metrics. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of the proof of the main theorem.

Lars Louder - Negative immersions for one-relator groups
Abstract: We prove a freeness theorem for low-rank subgroups of one-relator groups. The proof generalizes several classical facts about free and one-relator groups, including: Lyndon's theorem that a commutator in a free group is not a proper power, Magnus' Freiheitssatz, Wise's W-cycles conjecture, Stallings' theorem that injections of free groups inducing injections on abelianization are injective on the set of conjugacy classes, and Baumslag's theorem on adjoining roots to subgroups of free groups. The simplest new case covered resembles the Borromean rings. This is joint work with Henry Wilton.

Lee Mosher - Stable translation lengths on the free splitting complex (joint with Michael Handel)
Abstract: We prove that under the action of Out(F_n) on the free splitting complex of F_n, there is a positive lower bound to the stable translation length of any loxodromic element. The proof uses interactions between relative train track maps and free splitting units.

Zlil Sela - Basic conjectures and preliminary results in non-commutative algebraic geometry
Abstract: Algebraic geometry stdies the structure of varieties over fields and commutative rings. Starting in the 1960's ring theorists (Cohn, Bergman and others) have tried to study the structure of varieties over some non-commutative rings (notably free associative algebras).
The lack of unique factorization that they tackled and studied in detail, and the pathologies that they were aware of, prevented any attempt to prove or even speculate what can be the properties of such varieties.
Using techniques and concepts from geometric group theory and from low dimensional topology, we formulate concrete conjectures about the structure of these varieties, and prove preliminary results in the direction of these conjectures.

Jing Tao - Genus bounds in right-angled Artin groups
Abstract: The commutator length and stable commutator length (scl) are important quantities which are closely related to bounded cohomology, knot theory and 3-manifolds, yet the computation and estimation of such quantities are usually difficult, even in the case of free groups. I will describe an elementary topological argument which gives lower bounds for stable commutator lengths in right-angled Artin groups.

Karen Vogtmann - Feynman amplitudes and Outer space
Abstract: Recent work of Bloch and Kreimer has indicated that certain rules used by physicists for computing Feynman amplitudes in quantum field theories are related to the structure of Outer space, in particular to the combinatorics of both the spine and the Bestvina-Feighn bordification. I will attempt to explain this relationship.