Topology & Group Theory Seminar

Vanderbilt University


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(Fn)

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(Fn). This is joint work with Vincent Guirardel and Camille Horbez.

Wednesday, October 11, 2017

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 P1. We will show some pictures of Julia and Fatou sets of these maps.

Wednesday, October 18, 2017

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, at 5:10 pm

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 g1,...,gk are independent, atoroidal, fully irreducible outer automorphisms of the free group Fn, then there is a power m so that subgroup generated by g1m,...,gkm gives rise to a hyperbolic extension of Fn with rank n+k. Joint work with Sam Taylor.

Wednesday, December 6, 2017, at 5:10 pm

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, January 17, 2018 at 5:10 pm.

Yair Hartman (Northwestern University)

Title: Stationary C*-dynamical systems

Abstract: We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact. Joint work with Mehrdad Kalantar.

Wednesday, January 24, 2018

Alexander Olshanskii (Vanderbilt)

Title: Polynomially-bounded Dehn functions of groups

Abstract: The minimal non-decreasing function d(n): ℕ → ℕ such that every word w vanishing in a group G= < A | R > and having length ||w|| ≤ n is freely equal to a product ≤ d(n) conjugates of relators from R or R-1 , is called the Dehn function of the presentation < A | R > . In other words, the Dehn function d(n) of the presentation is the smallest function that bounds from above the areas of loops of length at most n in the Cayley complex Cay(G).

On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions d(n) with growth at least n4 (essentially all possible such Dehn functions) constructed by M. V. Sapir, J. C. Birget and E. Rips, in [Isoperimetric and isodiametric functions of groups, Annals of Mathematics, 157, 2(2002), 345-466] and based on the time functions of Turing machines and S-machines. The class of Dehn functions n a with a in (2, 4) remained more mysterious even though it has attracted quite a bit of attention (for example, [N. Brady and M. Bridson, There is only one gap in the isoperimetric spectrum, Geometric and Functional Analysis, 10 (2000), 1053-1070]). We fill the gap obtaining Dehn functions of the form na (and much more) for all real a > 2 computable in reasonable time, for example, a = π or a = e, or a any algebraic number [A.Yu. Olshanskii, Polynomialy-bounded Dehn functions of groups, arXiv: 1710.00550, 2017, 92 pp.]

Wednesday, January 31, 2018

Yash Lodha (EPFL in Lausanne)

Title: Rigidity of coherent actions by homeomorphisms of the real line and subgroups of Thompson's group F.

Abstract: We study actions of groups by homeomorphisms on (or the closed interval [0, 1]) that are minimal, have solvable germs at ∞ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. As a first application, we establish that the Brown-Stein groups F (2, p1, ..., pn) for n ≥ 1 and p1, ..., pn distinct odd primes, do not embed into Thompsons group F. This answers the following question of M. Brin: Does every finitely generated subgroup of the group of orientation preserving piecewise linear homeomorphisms of [0,1] embed into Thompsons group F? As a second application, we demonstrate that any coherent group action G < Homeo+(ℝ) that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompsons group F. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms.

Wednesday, February 7, 2018

Viveka Erlandsson (University of Bristol, UK)

Title: Geodesic Currents and Counting Curves

Abstract: Due to Mirzakhani we know the asymptotic growth of the number of curves, with bounded self-intersection, on a hyperbolic surface of bounded length L, as L grows. If S is a surface of genus g equipped with a hyperbolic structure, she showed that the number of such curves on S (in each mapping class group orbit) is asymptotic to a constant times L6g-6. In this talk I will explain, through the use of geodesic currents, why the same asymptotics hold for other metrics on the underlying topological surface.

Wednesday, February 14, 2018

Anthony Genevois (Aix-Marseille University, France)

Title: Quasi-median graphs and group theory

Abstract: The goal of this talk is to give an overview of the applications of quasi-median graphs, a class of graphs generalising CAT(0) cube complexes, to geometric group theory. Groups which will be considered include graph products, some lamplighter groups, and Thompson's groups.

Wednesday, February 28, 2018

Ashot Minasyan (Southampton, UK)

Title: Conjugacy separability of non-positively curved groups

Abstract: A group G is said to be conjugacy separable if given two non-conjugate elements x,y ∈ G, there is a homomorphism ψ from G to a finite group M such that ψ(x) and ψ(y) are not conjugate. Conjugacy separability, along with residual finiteness, are two classical residual properties of groups, which measure how well a given infinite group can be approximated by its finite quotients. These properties can be viewed as algebraic analogues of solvability of the conjugacy problem and the word problem in the group respectively. For many groups residual finiteness can be shown quite easily (e.g., all finitely generated linear groups are residually finite). Conjugacy separability, on the other hand, is much harder to prove, and until recently only a few classes of conjugacy separable group were known. During the talk I will survey some recent tools and progress in establishing conjugacy separability of "non-positively curved" groups.

Wednesday, March 7, 2018
No seminar: Spring break.

Wednesday, March 21, 2018

Dan Farley (Miami University)

Title: Quasi-automorphism groups of type F

Abstract: In this talk, I will consider certain certain groups QF, QT, and QV that act as quasi-automorphisms on the infinite binary tree. These groups have similar definitions to the well-known Thompson groups F, T, and V (respectively). I will show that all three quasi-automorphism groups act properly on infinite dimensional CAT(0) cubical complexes, and have type F. The proof features an application of Brown's well-known finiteness criterion. Nucinkis and St. John-Green proved similar F results for QF and QV by different methods; the QT case considered here answers a question from their work.

The CAT(0) cubical complexes for QF, QT, and QV are built using Guba and Sapir's theory of diagram groups. Guba and Sapir introduced [standard] diagram groups, annular diagram groups, and braided diagram groups. The groups QF, QT, and QV are hybrids of these types (as I will describe in the talk).

The method admits several generalizations, which will be considered if time permits.

This is joint work with Delaney Aydel and Samuel Audino.

Wednesday, March 28, 2018

Denis Osin (Vanderbilt)

Title: Acylindrically hyperbolic groups with exotic properties

Abstract: My talk will be based on a join work in progress with A. Minasyan. We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group Q such that Q has property FLp for all p>1 and every action of Q on a finite dimensional contractible topological space has a fixed point. In addition, Q is not uniformly non-amenable and has finite generating sets with arbitrary large balls consisting of torsion elements.

Wednesday, April 11, 2018

Ivan Levcovitz (CUNY, New York)

Title: Coarse geometry of right-angled Coxeter groups

Abstract: A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a brief background of what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.