Topology & Group Theory Seminar

Vanderbilt University

2016/2017

Organizer: Mark Sapir

Wednesdays, 4:10pm in SC 1310 (unless otherwise noted)

** Wednesday, January 20, 2016 **

Andrew Sale (Vanderbilt)

Title: The permutation conjugacy length function

Abstract: I will describe a geometric version of the conjugacy problem, which involves determining the length of short conjugators between elements in a group, and defining the conjugacy length function. I will then explain recent work with Y. Antoln in which we define the permutation conjugacy length function, a similar but more delicate notion, which has the potential to form a base for a geometric study of the conjugacy problem that yields fast algorithms solving the conjugacy search problem.

** Wednesday, January 27, 2016 **

Caglar Uyanik (UIUC)

Title: Dynamics of free group automorphisms and a subgroup alternative for Out(F_{N})

Abstract: The study of outer automorphism group of a free group Out(F_{N}) is closely related to the study of Mapping Class Group of a surface. We will discuss various free group analogs of pseudo-Anosov homeomorphisms of hyperbolic surfaces. We will focus mostly on dynamics of their actions on the space of currents and deduce several structural results about subgroups of Out(F_{N}). Part of this talk is based on joint work with Matt Clay.

** Wednesday, February 3, 2016 **

John Ratcliffe (Vanderbilt)

Title: Salem numbers and arithmetic hyperbolic groups.

Abstract: This is joint work with Vincent Emery and Steven Tschantz. In this paper, we prove that there is a direct relationship between Salem numbers and translation lengths
of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field.

** Wednesday, February 10, 2016 **

Yuri Bahturin (Memorial University of Newfoundland)

Title: Real Graded Division Algebras

Abstract: This is joint work with Mikhail Zaicev of Moscow State University. Classifying real graded division algebra is a necessary step in the classification of group gradings on simple real Lie and Jordan algebras. We approach this classification using algebras given in terms of generators and defining relations. An important class of real division graded algebras is provided by canonical gradings on Clifford algebras and Pauli/Sylvester gradings on complex matrix algebras.

** Wednesday, February 17, 2016 **

Volodymyr Nekrashevych (Texas A&M)

Title: Periodic groups from minimal actions of the dihedral group

Abstract: We will give a construction transforming an arbitrary minimal non-free
action of the infinite dihedral group on the Cantor set into an
orbit-equivalent action
of a finitely generated amenable periodic group. In particular, we
construct first examples of simple infinite finitely generated
amenable periodic groups. Some explicit examples: groups of polygonal
rearrangements, groups associated with circle rotations, etc., will be
discussed.

** Wednesday, February 24, 2016 **

Yunxiang Ren (Vanderbilt)

Title: Skein theory for subfactors and presentations for Thompson group

Abstract: By Vaughan Jones' work, a lot of representations of Thompson group could be constructed
with the data of subfactors and some interesting subgroups could be arise from this way. In this
talk, we will reveal the relation between skein theory for subfactors and presentation for those
subgroups obtained from the subfactors.

** Wednesday, March 2, 2016 **

Sam Corson (Vanderbilt)

Title: Some new results about n-slender groups

Abstract: Groups which canonically have an infinitary multiplication structure have few homomorphisms to certain other groups. I will first discuss slender groups and give motivating examples. I will then discuss n-slender groups and a recent theorem that torsion-free word hyperbolic groups are n-slender.

** Wednesday, March 16, 2016 **

Spencer Dowdall (Vanderbilt)

Title: The geometry of hyperbolic free group extensions

Abstract: Each subgroup Γ of the outer automorphism group of a free group naturally gives rise to a group extension E_{Γ} of the free group. In joint work with Sam Taylor, we have given geometric conditions on Γ that imply the extension is a hyperbolic group. After describing these conditions, I will present recent results about the fine geometric structure of these group extensions. These results include a Scott--Swarup type theorem proving quasiconvexity of infinite-index subgroups of the free group, a quantitative description of the Cannon--Thurston map of the extension, and a width theorem that characterizes these hyperbolic extensions in terms of the axes of primitive elements of the free group.

** Wednesday, March 23, 2016 **

Sahana Hassan Balasubramanya (Vanderbilt)

Title: Acylindrical group actions on quasi-trees

Abstract: A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph is a (non-elementary) quasi-tree and the action of G on the Cayley graph is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As a by-product, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.

** Wednesday, March 30, 2016 **

Mike Mihalik (Vanderbilt)

Title: Coaxial Semistability for Spaces

Abstract: The history of this work begins in 1935 with a famous example of J.H.C.
Whitehead, of a contractible open 3-manifold not homeomorphic to Euclidean 3-
space. In 1988, R. Meyers showed this manifold and a large collection of analogous
manifolds (due to D. R. McMillan- 1962) are not (non-trivial) covering spaces. We
give a brief history of how D. Wright generalized Meyers work in 1992 and how R.
Geoghegan and C. Guilbaugh generalized Wrights work in 2015 to a seemingly
capstone result about infinite cyclic group actions on simply connected spaces that
have pro-monomorphic fundamental group at infinity.
Finally we will state and outline a proof of a result that generalizes the
Geoghegan/Guilbaugh result by replacing infinite cyclic group actions by finitely
generated group actions and replacing the pro-monomorphic hypothesis by a much
weaker geometric hypothesis. This is joint work with R. Geoghegan and C. Guilbault.

** Wednesday, April 6, 2016 **

Vyacheslav Kharlamov (University of Strasbourg)

Title: Real cubic projectives hypersurfaces

Abstract: Cubic hypersurfaces is one of the classical objects of study in real algebraic geometry. While the case of cubic surfaces can be considered as rather well understood, the case of cubics of dimensions five and higher remain still largely open (even over the complex field). The topological and deformation classifications of real cubic hypersurfaces in dimensions 3 and 4 was achieved only recently. In a joint work with S. Finashin (work in progress) we suggest a solution of one further, related, problem, that of the topological and deformation classifications of pairs consisting of a real cubic 3-fold and a real straight line
contained in it. Our approach is based on a certain, spectral, correspondance between such pairs and plane quintics equipped with a real theta-characteristic. This correspondance allows to disclose some new phenomena both on the cubic and quintic sides.

** Wednesday, April 13, 2016 **

Andrew Stewart (University of Toronto)

Title: Scaling limits of random walk bridges on trees

Abstract: Let X_{n} be a nearest neighbour random walk on either the free group F_{k} of rank at least 2 or the free product of at least 3 copies of the cyclic group of order 2. Let T_{N} be the subtree visited by X_{n}, conditioned on X_{N} = X_{0}. By proving a recursive characterization of the Brownian continuum random tree, we show that ^{1}⁄_{
√ N }
T_{N} converges in distribution to an appropriately rescaled Brownian continuum random tree.