Topology & Group Theory Seminar

Vanderbilt University

2013/2014

Organizer: Mark Sapir

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

** Monday 23 August 4:10-5:30 inm SC 1432 4:10-5:30 inm SC 1432 4:10-5:30 inm SC 1432 4:10-5:30 inm SC 1432 (joint with the Subfactors seminar)**

Speaker: Cary Malkiewich (Stanford)

Title:
Coassembly Maps, Gauge Groups, and K-Theory

Abstract:Calculus of functors is a powerful technique from homotopy theory, which studies computationally difficult constructions by means of "linear approximations." We will describe a new variant of this calculus, based on the embeddingÂ calculusÂ of Weiss and Goodwillie. This theory provides us with a sequence of approximations to the stable gauge group of a principal bundle, in which the linear approximation is the Cohen-Jones string topology spectrum. We will finish with some future applications to algebraic K-theory, a subject which provides a powerful (but difficult to compute) invariant for rings, algebras, and groups.

** Wednesday, 28 August 2013. **

Speaker: Jenya Sapir (Stanford)

Title: Counting Non-Simple Closed Geodesics on a Pair of Pants

Abtsract: We give coarse bounds on the number of (non-simple) closed geodesics on a pair of pants with upper bounds on both length and intersection number. We acheive our bounds by transforming the problem of counting geodesics into a combinatorial problem of counting words with certain conditions.
If time permits, we will give an idea of how to extend these results to a general surface, and give an application.

** Wednesday, 4 September 2013 **

Speaker: Alexander Olshanskii (Vanderbilt University)

Title: On Pairs of Finitely Generated Subgroups in Free Groups

Abstract:We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F,
there is a subgroup H of finite index in B such that the subgroup < A,H > generated by A and H has infinite index in F.
The main corollary of this theorem says that a free group of free rank r > 1 admits a faithful highly transitive action,
whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.

** Wednesday, 11 September 2013 **

Speaker: Nathaniel Pappas (University of Virginia)

Title: Values of Rank Gradient and p-Gradient

Abstract: The rank gradient and p-gradient are group invariants which assign some real number greater than or equal to -1 to a finitely generated group. Mark Lackenby first defined rank gradient and p-gradient as means to study 3-manifold groups. Rank gradient has connections with other group invariants from other fields of mathematics such as cost and L2 Betti numbers. The question of what values can be obtained as the rank gradient of some finitely generated group has remained open. I will discuss the related problem of determining what values can be achieved by the p-gradient as well as how to compute rank gradient and p-gradient of free products with amalgamation over an amenable subgroup and HNN extensions with amenable associated subgroup.

** Wednesday, 18 September 2013 **

Speaker: Yago Antolin Pichel (Vanderbilt)

Title: Relatively hyperbolic groups with the falsification by fellow traveler property

Abstract: The falsification by fellow traveler property is a property of the Cayley graph of a group. It was introduced by Neumann and Shapiro, it has several implications. For example, it implies the regularity of the language of geodesics, the rationality of the growth series or having a quadratic Dehn function. I will explain how to find a generating set with the falsification by fellow traveler property for groups relatively hyperbolic to groups with a generating set with this property. This is a joint work with Laura Ciobanu.

** Wednesday, 2 October 2013**

Speaker: Dennis Dreesen (University of Southampton)

Title: Locally compact hyperbolic groups

Abstract: The common convention when dealing with hyperbolic groups is that such groups are finitely generated and equipped with the word length metric relative to a finite symmetric generating subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the finitely generated setting. We study the class of locally compact hyperbolic groups and elaborate on the similarities and differences between the discrete and non-discrete setting.

** Wednesday, 9 October 2013**

Speaker: Dmitry Burago (Pennstate)

Title: "Stories from another pocket" (after Karel Capek and others).

Abstract: This year, I have been delivering a number of talks under almost the same title.
However, the talks are quite different. I have prepared about twenty topics, with
twothree slides for each. For each talk, I select about eight topics, the choice
depends on the audience, how long the talk is etc. The topics are united only by
the fact they were of interest to me in the past several years.
For each topic, I give only key definitions, one or two theorems and several
open problems (which may form the most important part of the talk). The talk
is supposed to be accessible to (reasonable) graduate students. We will not
go into (almost:) any technicalities.

** Wednesday, 16 October 2013 **

Speaker: Tsachik Gelander (Hebrew University and Weizmann Institute)

Title: Lattices in Amenable Groups.

Abstract: Let G be a locally compact amenable group. We discuss the question whether every closed subgroup of finite covolume in G is cocompact. Joint work with U. Bader, P.E. Caprace and S. Mozes.

** Wednesday, 23 October 2013 **

Speaker: Rémi Coulon (Vanderbilt)

Title: Groups with a tree-graded asymptotic cone.

Abstract: Given a finitely generated group G, its asymptotic cones is a class of objects that capture the large scale properties of the group.
Roughly speaking they are obtained by looking at G from infinitely far away.
The asymptotic cones of a hyperbolic group are trees.
Conversely if G is a finitely presented group such that some asymptotic cone of G is a tree, then G is hyperbolic.
The goal of this talk is to present a generalization of this statement for the class of relatively hyperbolic groups.
This is a joint work with M. Hull and C. Kent.

** Wednesday, 30 October 2013 **

Speaker: Darren Creutz (Vanderbilt)

Title: Character Rigidity for Lattices and Commensurators

Abstract: Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, answering a question of Connes and generalizing our result that every nonatomic probability-preserving action of such a group is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it into the operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

** Wednesday, 13 November 2013 **

Speaker: Laura Ciobanu (University of Neuchatel)

Title: Conjugacy languages and growth series in groups

Abstract: In this talk I will introduce two languages related to conjugacy in groups, and discuss their regularity in lots of classes of groups, including hyperbolic, virtually abelian, Artin and more.
I will also present results regarding the conjugacy growth series in free products and graph products, and show that the conjugacy growth series of a virtually cyclic group is rational for all generating sets.
This is joint work with Susan Hermiller, Derek Holt, and Sarah Rees.

** Wednesday, 20 November 2013 **

Speaker: Rares Rasdeaconu (Vanderbilt)

Title: Counting real curves on K3 surfaces

Abstract: Real enumerative invariants of real algebraic manifolds are derived from counting real curves with suitable signs.
I will discuss the case of counting real rational curves on simply connected complex projective surfaces with zero first
Chern class (K3 surfaces) equipped with an anti-holomorphic involution. An adaptation to the real setting of a formula
due to Yau and Zaslow will be presented. The proof passes through results of independent interest: a new insight into
the signed counting, and a formula computing the Euler characteristic of the real Hilbert scheme of points on a K3 surface,
the real version of a result due to Gottsche.
The talk is based on a joint work with V. Kharlamov.

** Wednesday, 27 November 2013 **

No seminar: Thanksgiving break in Vanderbilt.

** Wednesday, 4 December 2013, 2:10 - 3:00 (same room)**

Speaker: Yash Lodha (Cornell University)

Title: A geometric approach to the von Neumann problem

Abstract: We construct a finitely generated group of piecewise projective transformations of the circle which is finitely presented, has no no-abelian free subgroups and is not amenable. This is a joint work with Justin Moore.

** Wednesday, 4 December 2013 **

Speaker: Anton Malyshev (UCLA)

Title: Growth and nonamenability in product replacement graphs

Abstract: The product replacement graph (PRG) of a group $G$ is the
set of generating $k$-tuples of $G$, with edges corresponding to
Nielsen moves. It is conjectured that PRGs of infinite groups are
nonamenable. We verify that PRGs have exponential growth when $G$ has
polynomial growth or exponential growth, and show that this also holds
for a group of intermediate growth: the Grigorchuk group. We also
provide some sufficient conditions for nonamenability of the PRG,
which cover elementary amenable groups, linear groups, and hyperbolic
groups.

** Wednesday, 15 January 2014 **

Speaker: Mark Sapir (Vanderbilt University)

Title: The Tarski numbers of groups

Abstract: The Tarski number of a non-amenable group is the minimal number of pieces in a paradoxical decomposition of the group. It is known that a group has Tarski number 4 if and only if it contains a free non-cyclic subgroup, and the Tarski numbers of torsion groups are at least 6. It was not known whether the set of Tarski numbers is infinite and whether any particular number >4 is the Tarski number of a group. We prove that the set of possible Tarski numbers is infinite even for 2-generated groups with property (T), show that 6 is the Tarski number of a group (in fact of any group with large enough first L_2-Betti number), and prove several results showing how the Tarski number behaves under extensions of groups. This is a joint work with Mikhail Ershov and Gili Golan.

** Wednesday, 22 January 2013 **

Speaker: Yago Antolin Pichel (Vanderbilt)

Title: Residual finiteness of some outer automorphism groups.

Abstract: In this talk I will present my work together with Minasyan and Sisto on commensurating endomorphism of acylindrically hyperbolic groups. We show that any commensurating endomorphism of an acylindrically hyperbolic groups is inner modulo a small perturbation. We use this result to prove that the outer automorphism group of a virtually special group or the fundamental group of some compact 3-manifold is residually finite.

** Wednesday, 12 February 2014 **

Speaker: Vincent Guirardel (University of Rennes 1)

Title:Finite groups of automorphisms and finite extensions of relatively hyperbolic groups

Abstract: We study finite subgroups of Out(G) when G is hyperbolic relative to virtually polycyclic groups, and we prove that there are only finitely many conjugacy classes of them. As an application, we give some finiteness results concerning finite extensions of G.
This is a joint work with Gilbert Levitt.

**Wednesday, 19 February 2014, 3 pm, same room **

Speaker: Anatoly Vershik (St Petersburg, Russia)

Title: Invariant measures and intrinsic metric on the Bratteli diagrams

Abstract: How to find the list of the traces of an AF-algebra (characters of the locally finite group). The theory of filtrations (decreasing
sequencs of sigma-fields), which was developed years ago in the
framework of measure theory and ergodic thery, gave a natural method
for distiguish AF-algebras.

** Wednesday, 19 February 2014**

Speaker: Denis Osin (Vanderbilt)

Title: C*-simple groups without free subgroups.

Abstract: We construct first examples of non-trivial groups without non-cyclic free subgroups whose reduced C*-algebra is simple and has unique trace. This answers a question of de la Harpe. Both torsion and torsion free examples are provided. In particular, we show that the reduced C*-algebra of the free Burnside group B(m,n) of rank m> 1 and any sufficiently large odd exponent n is simple and has unique trace.

** Wednesday, 26 February 2014 (joint with the Subfactors seminar)**

Speaker: Alex Furman (UIC)

Title: Classifying lattice envelopes of some countable groups.

Abstract: In a joint work with Uri Bader and Roman Sauer we study the following question:
given a countable group L describe all locally compact groups G which
contain a copy of L as a lattice (uniform or non-uniform).
I will discuss the solution of this problem for a large class of
countable groups L.
The proof involves a somewhat unexpected mix of tools, including:
Breuillard-Gelander's topological Tits alternative, Margulis'
commensurator superrigidity, arithmeticity, and normal subgroup
theorems, quasi-isometric rigidity results of Kleiner-Leeb, and
Mosher-Sageev-Whyte.

**Wednesday, April 2 2014**

Speaker: Aditi Kar (Oxford)

Title: Gradients in Group Theory

Abstract: Rank and Deficiency gradients quantify the asymptotics of finite approximations of a group. These group invariants have surprising connections with many different areas of mathematics: 3-manifolds, L2 Betti numbers, topological dynamics and profinite groups.
I will give a survey of the current state of research in Gradients for groups and describe important open questions. Joint work with Nikolay Nikolov.

** Wednesday, April 9 2014. 3:10 pm, same room. **

Speaker: Romain Tessera (Universite Paris-Sud)

Title: A complete invariant for generalized Baumslag-Solitar groups

Abstract: Most of the time, proving that two groups are quasi-isometric amounts to showing that they both act properly and cocompactly by isometries on a same metric space. This defines a relation between locally compact groups, which we can extend to a natural equivalence relation named "commability". Commability lies somewhere between being quasi-isometric and being commensurable. In this talk, we will provide a simple, computable, complete commability invariant for generalized Baumslag-Solitar groups. This is in sharp contrast with the fact that the classification of such groups up to commensurability is a wide open problem.

**Wednesday, April 16 2014 **

Speaker: John Ratcliffe (Vanderbilt)

Title: On the isometry group of a compact flat orbifold

Abstract: Let Γ be an n-dimensional crystallographic group.
We prove that the group Isom ( E^{n}/Γ) of isometries of
the flat orbifold E^{n}/Γ is a compact Lie group
whose component of the identity is a torus of dimension equal to the
first Betti number of the group Γ.
This implies that Isom(E^{n}/Γ) is finite if and only if
Γ/[Γ, Γ] is finite.
We also generalize known results on the Nielsen realization problem for torsion-free Γ to arbitrary Γ.