Yael Algom-Kfir - The boundary of a hyperbolic free-by-cyclic group
Abstract: Given an automorphism \phi of the free group Fn consider
the HNN extension Gφ = Fn \rtimes Z. We compare two cases:
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.
- 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).
- φ is atoroidal and fully irreducible. Then by a theorem of
Brinkmann Gφ is hyperbolic. If φ is irreducible
then its boundary is homeomorphic to the Menger curve (M. Kapovich
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
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 Fn
Abstract: This is joint work with Mark Feighn. I will discuss our solution to the conjugacy problem for rotationless polynomially growing outer automorphisms of Fn.
Camille Horbez - Abstract commensurators of some normal subgroups of
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
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(Fn) on the free splitting complex of Fn, 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
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