Topology & Group Theory Seminar

Vanderbilt University

2017/2018

Organizer: Mark Sapir

Wednesdays, 4:10pm in SC 1308 (unless otherwise noted)

** Wednesday, August 30, 2017 **

Mike Mihalik (Vanderbilt)

Title: Semistability of relatively hyperbolic groups.

Abstract:
Suppose *G * is a 1-ended finitely presented group that is hyperbolic relative to ** P ** a finite collection of 1-ended finitely generated subgroups. Our main theorem states that if

∂ (G, ** P **) has no cut point, then *G * has semistable fundamental group at ∞. Under mild conditions on *G* and the members of ** P** the 1-ended hypotheses and the no cut point condition can be eliminated to obtain the same semistability conclusion.

** Wednesday, September 6, 2017 **

Matthew Haulmark (Vanderbilt)

Title: A classification theorem for 1-dimensional boundaries of groups with isolated flats.

Abstract: In 2000 Kapovich and Kleiner proved that if G is a one-ended hyperbolic group that does not split over a two-ended subgroup, then the boundary of G is either a Menger curve, a Sierpinski carpet, or a circle. Kim Ruane observed that there were no known non-hyperbolic examples of groups with Menger curve boundary, and asked if there was a CAT(0) generalization of Kapovich and Kleiners theorem. As boundaries of CAT(0) groups are in general not locally connected, there is no hope of such a generalization for general CAT(0) groups. However, a version of Kapovich and Kleiners theorem may hold for certain classes of CAT(0) groups. In this talk I will discuss a generalization of the Kapovich-Kleiner theorem for CAT(0) groups with isolated flats, and provide an example of a non-hyperbolic CAT(0) group with Menger curve boundary.

**Wednesday, September 13, 2017**

Arman Darbinyan (Vanderbilt)

Title: Word and conjugacy problems in finitely generated groups

Abstract: We will discuss some new results about the relationship between word and conjugacy problems in finitely generated groups.
In particular, we will discuss a method which allows us to construct: (1) finitely generated (solvable or finitely presented) torsion-free groups with
decidable word problem and such that they cannot be embedded into groups with decidable conjugacy problem;
(2) finitely generated groups with decidable semi-conjugacy problem and undecidable conjugacy problem.
Both (1) and (2) answer questions which were known as open questions.

** Wednesday, September 27, 2017**

Caglar Uyanik (Vanderbilt)

Title: Dynamics and geometry of free group automorphisms

Abstract: I will talk about the long standing analogy between the mapping class group of a hyperbolic surface and the outer automorphism group of a free group. Particular emphasis will be on the dynamics of individual elements and applications of these results to structural theorems about subgroups of these groups.

** Wednesday, October 4, 2017 **

Mladen Bestvina (University of Utah)

Title: Boundary amenability of *Out(F _{n})*

Abstract: I will discuss boundary amenability and how to prove it for
basic groups for most of the hour. The main interest in boundary
amenability is that it implies the Novikov conjecture in manifold
theory. I will then outline the main ideas in the proof of boundary
amenability of *Out(F _{n})*. This is joint work with Vincent Guirardel and
Camille Horbez.

Vaughan Jones (Vanderbilt)

Title: Coefficients of certain unitary representations of Thompson groups.

Abstract: There is a general construction, essentially due to Ore, of a group of quotients
of certain rather special categories. If the category is planar rooted binary forests (under stacking),
the group of quotients is Thompsons group *F*. Actions of these groups of fractions can be constructed
whenever there is a functor from the underlying category to another category that admits direct limits.
I will briefly review these constructions and show how to construct many actions of F, including
unitary representations. Irreducibility is then a major question. As a first attempt to tackle this problem
I will focus on the *coefficients* of these unitary representations. Calculation of coefficients
led to a construction of knots and links but we will see that it also leads to the study of iteration of
dynamical systems which in the simplest cases are rational functions on *ℂ P ^{1}*. We will
show some pictures of Julia and Fatou sets of these maps.

Guoliang Yu (Texas A & M)

Title: Non-rigidity of manifolds and secondary invariants of elliptic operators

Abstract: I will explain how to use certain secondary invariants of elliptic operators to measure degree of non-rigidity for manifolds and discuss the connection to group theory. This is joint work with Shmuel Weinberger and Zhizhang Xie.
I will try my best to make this talk accessible.

** Wednesday, October 25, 2017 **

Larry Rolen (Trinity College, Dublin and Georgia Tech)

Title: An overview of mock modular forms and their applications in combinatorics, geometry, algebra, and topology.

Abstract: Modular forms are central objects in contemporary mathematics. In the last few decades, modular forms have been featured in fantastic achievements such as progress on the Birch and Swinnerton-Dyer Conjecture, mirror symmetry, Monstrous moonshine, and the proof of Fermats Last Theorem. This talk is about a generalization of the theory of modular forms and the corresponding extension of their web of applications. We will begin by surveying the history of this subject as it arose from the enigmatic deathbed letter of Indian mathematical genius Ramanujan. After 80 years, the Ph.D. thesis of Zwegers finally "explained" the mystery of Ramanuan's functions. This has led to an explosion of activity in the field of mock modular forms in the last decade. Here, I will describe a sampling of the applications, in particular to partitions, ranks of elliptic curves, algebra, and knot theory. The talk will be accessible for a general audience; in particular, no knowledge of number theory or modular forms is assumed.

** Wednesday, November 1, 2017 **

Mark Sapir (Vanderbilt)

Title: Divergence functions of R. Thompson groups

Abstract: This is a joint work with Gili Golan. The divergence function of a group generated by a finite set *X* is the smallest function *f(n)* such that for every *n* every two elements of length *n* can be connected in the Cayley graph (corresponding to *X*) by a path of length at most *f(n)* avoiding the ball of radius *n/4* around the identity element. We prove that R. Thompson groups * F, T, V* have linear divergence functions. Therefore the asymptotic cones of these groups do not have cut points.

** Wednesday, November 8, 2017 **

Matt Clay (University of Arkansas)

Title: Thermodynamic metrics on outer space

Abstract: I will describe some ongoing work with Tarik Aougab and Yov Rieck towards understanding metrics on outer space arising from the tools of thermodynamic formalism.

** Wednesday, November 15, 2017 **

No seminar

** Wednesday, November 29, 2017 **

Spencer Dowdall (Vanderbilt)

Title: On ranks of hyperbolic group extensions.

Abstract: The rank of a group is the minimal cardinality of a generating set. While simple to define, this quantity is notoriously difficult to calculate, and often uncomputable, even for well behaved groups. In this talk I will explain general conditions that may be used to show many hyperbolic group extensions have rank equal to the rank of the kernel plus the rank of the quotient. In this case, we further show that any minimal generating set is Nielsen equivalent to one in a standard form. As an application, we prove that if *g _{1},...,g_{k}* are independent, atoroidal, fully irreducible outer automorphisms of the free group

Hung Cong Tran (University of Georgia)

Title: On strongly quasiconvex subgroups

Abstract: We introduce the concept of strongly quasiconvex subgroups of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We prove that strongly quasiconvex subgroups have many properties analogous to those of quasiconvex subgroups of hyperbolic groups. We study strong quasiconvexity and stability in relatively hyperbolic groups, two dimensional right-angled Coxeter groups, and right-angled Artin groups. We note that the result on right-angled Artin groups strengthens the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

**Wednesday, February 7, 2018 **

Viveka Erlandsson (University of Bristol, UK)

Title: TBA

Abstract: TBA

**Wednesday, February 14, 2018**

Anthony Genevois (Aix-Marseille University, France)

Title: TBA

Abstract: TBA