Topology & Group Theory Seminar

Vanderbilt University

2011/2012

Organizer: Mark Sapir

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

** Wednesday, 31 August 2011**

Speaker: Mike Mihalik (Vanderbilt University)

Title: Ends and quasi-normal subgroups of groups

Abstract:
We develop the theory of quasi-normal subgroups of groups in
analogy with that of normal subgroups. The subgroup Q of the group G is
quasi-normal in G if each element of G is in the commensurator of Q. Our
main result is: If a finitely generated group G contains an infinite
finitely generated quasi-normal subgroup of infinite index, then G is
1-ended and semistable at infinity (morally, G is geometrically connected
and locally connected at infinity). We explain how this result can be used
to generalizes nearly every ``normal type" semi-stability result in the
literature.

** Wednesday, 7 September, 2011 **

Speaker: Mark Sapir (Vanderbilt University)

Title: Aspherical groups and manifolds with extreme properties

Abstract: We prove that every aspherical recursively presented group
embeds into a group with finite aspherical presentation complex. By
results of Gromov and Davis, this implies that there exists a closed
aspherical manifold of any dimension >3 (smooth in dimension >4) with
universal cover of infinite asymptotic dimension, and which is a
counterexample to the Baum-Connes conjecture with coefficients.

** Wednesday, 21 September, 2011 **

Speaker: Michael Brandenbursky (Vanderbilt University)

Title: Quasi-isometric embeddings into diffeomorphism groups

Abstract: Let M be a smooth compact connected oriented manifold
of dimension at least two endowed with a volume form. Assuming
certain conditions on the fundamental group of M, we
construct quasi-isometric embeddings of either free Abelian or direct
products of non-Abelian free groups into the group of volume
preserving diffeomorphisms of M equipped with the L^{p} metric induced
by a Riemannian metric on M.

** Wednesday, 28 September, 2011 **

Speaker: Vahagn H. Mikaelyan (Yerevan State University)

Title: Normal and subnormal embeddings of groups

Abstract: Given a group H, is there a larger group G such that H is normal in G and H is contained in the commutator subgroup [G,G] ? We suggest a criterion answering this and more
general problem of Heineken. An embedding of a group H into a group G is normal (subnormal) if the image of H is a normal (subnormal) subgroup in G. And for the non-trivial word set V the embedding is verbal if the image of H lies in the verbal subgroup V(G). We prove that there exists a group G admitting a normal verbal embedding of H into G if and only if V(Aut H) contains Inn H. For any group H and any non-trivial word set V there always exists a group G admitting a subnormal verbal embedding of H into G. In particular, if H is countable, then G can be chosen to be 2-generator. This is a generalization of the well-know theorem of Higman, Neumann and Neumann on embedding of any countable group into a 2-generator group.

** Wednesday, 12 October, 2011 **

Speaker: Tom Farrell (SUNY Binghamton)

Title: Bundles with negatively curved fibres

Abstract:
I will talk on joint work with Pedro Ontaneda. Let M be a closed smooth manifold which supports a negatively curved
Riemannian metric. And let p: E to B be a smooth bundle with abstract fibre M and family of fibres
E_x= p^{-1}(x), x in B. I will talk about the problem of equipping the fibres with negatively curved Riemannian
metrics varying continuously with x

** Wednesday, 26 October, 2011 **

Speaker: Rufus Willett (Vanderbilt University)

Title: Coarse geometry of expanding graphs, and graphs with large girth

Abstract: I will discuss expanding ('highly connected, low density') graphs, and graphs with large girth ('no small loops') from the point of view of large-scale geometry. Gromov (and Sapir) have shown that some such sequences of graphs 'embed' into (aspherical) groups, leading to several surprising results. Some of these properties are quite pathological, due to the complicated geometry of expanders, and lead to counterexamples to Baum-Connes type conjectures (among other things).
The two classes of graphs in the title share many similar properties: for example, graphs with large girth are expanders 'at small scales'. However, graphs (even expanders) with large girth are in some sense relatively 'benign' from the point of view of large scale geometry, partly as they do not have a 'geometric' version of property (T). I will survey all this. Much of what I will talk about is joint work with Guoliang Yu.

** Wednesday, 2 November, 2011 **

Speaker: Francois Dahmani (Grenoble, France)

Title: Groups of interval exchange transformations

Abstract:
(joint with V. Guirardel, K. Fujiwara) An interval exchange transformation is a bijective transformation of an interval that consists in cutting it into finitely many subintervals, and rearranging them by translations. These transformations have been much studied as individual dynamical systems. However, not much is known about the way they interact with each other. For this, it seems
natural to ask what kind of group two such transformations generate in the full group of interval exchange transformations. Is it possible to generate a free group, or an interesting solvable group ?
It turns out that free groups are "rare", and that torsion-free solvable groups are (virtually) abelian. However, there are lamplighter groups -- but lamps are necessarily abelian -- and uncountably many isomorphism classes of solvable groups.

** Wednesday, 9 November, 2011 **

Speaker: Darren Creutz (Vanderbilt University)

Title: Normal Subgroups of Commensurators of Lattices

Abstract: I will present some results of myself and Y. Shalom. I will focus on
our Normal Subgroup Theorem for Commensurators of lattices: any normal
subgroup of a (dense) commensurator of a lattice in a locally compact
group necessarily contains the lattice. Consequences of this theorem
will also be discussed: classification of normal subgroups of
commensurators; an improved form of Bader-Shalom's normal subgroup
theorem for lattices in products; and a partial answer to a question
of Lubotzky, Mozes and Zimmer on tree automorphisms.

** Wednesday, 16 November, 2011 **

Speaker: Igor Mineyev (UIUC)

Title: The Hanna Neumann Conjecture and generalizations.

Abstract: We will present a proof of the Strengthened Hanna Neumann Conjecture
(SHNC), and more general results. We will mention trees, forests, flowers,
gardens, and leafages.
Submultiplicativity is a generalization of the statement of SHNC from
graphs to complexes, and from free groups to more general groups.
Submultiplicativity holds for complexes under an additional assumption:
the deep-fall property.

** Wednesday, 30 November, 2011 **

Speaker: Darren Creutz (Vanderbilt University)

Title: SAT Actions and Rigidity of Lattices

Abstract: I will present an overview of SAT actions, a type of quasi-invariant
group action on a probability space that is the opposite of
measure-preserving, and recent work of Y. Shalom and myself on the
rigidity of such actions for lattices in the form of our SAT Factor
Theorem. I will then explain how this result plays the key role in
the previously presented work on Normal Subgroups of Commensurators of
Lattices.

** Wednesday, 7 December, 2011 **

Speaker: Denis Osin (Vanderbilt University)

Title: Inner non-amenability and simplicity of C*-algebras of groups
acting on hyperbolic spaces

Abstract:
I will discuss some general properties of groups acting
acylyndrically and non-elementary on hyperbolic spaces. In particular,
I will prove that the following conditions are equivalent for every
such a group G:
a) G has no non-trivial finite normal subgroups.
b) G is ICC (i.e., every nontrivial conjugacy class in G is infinite)
c) G is not inner amenable.
d) the reduced C*-algebra of G is simple with unique trace.

** Wednesday, 11 January, 2012 **

Speaker: Alexei Myasnikov (Stevens Tech)

Title: Groups acting freely on Lambda-trees and Z^{n}-hyperbolic groups

Abstract: TBA

** Wednesday, 18 January, 2012 **

Speaker: Olga Kharlampovich (Hunter College, New York)

Title: First order properties and algebraic geometry in groups in the presence of negative curvature

Abstract: I will do a survey of the subject.

** Wednesday, 25 January, 2012 **

Speaker: Mark Sapir (Vanderbilt University)

Title: Complicated residually finite groups.

Abstract: It is well known that the word problem in a finitely presented residually finite group is decidable. I will show that it can be arbitrarily complicated.

** Wednesday, February 1, 2012 **

Speaker: Greg Gauthier (Vanderbilt University)

Title: The Abelian sandpile model on infinite graphs

Abstract: I will discuss the sandpile model on the standard Cayley graph of the modular group. In particular I will show how to compute the critical exponent in that case.

** Wednesday, 8 February, 2012 **

Speaker: Harold Sultan (Columbia University)

Title: Asymptotic Geometry of Teichmuller Space and Divergence

Abstract: I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by
Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.

** Wednesday, 15 February, 2012 **

Speaker: Strom Borman (University of Chicago)

Title:
A geometric functionality for quasi-morphisms on the group of Hamiltonian diffeomorphisms

Abstract:
Quasi-morphisms on the group of Hamiltonian diffeomorphisms are a convenient way to package and see various rigidity phenomenon in symplectic topology. Their general construction due to Entov and Polterovich uses Hamiltonian Floer homology and requires a certain computation in the quantum homology ring. In this talk I will explain how symplectic reduction provides a sort of geometric functoriality that allows quasi-morphisms to descend along symplectic reductions without further quantum homology computations. The proof involves a lemma about pulling quasi-morphisms back along `quasi-homomorphisms. As an application, I will present a family of closed symplectic manifolds whose group of Hamiltonian diffeomorphisms have infinite dimensional spaces of quasi-morphisms .

** Wednesday, 22 February, 2012 **

Speaker: Curt Kent (Vanderbilt University)

Title: Understanding fundamental groups of asymptotic cones.

Abstract: Gromov was first to notice a connection between the homotopic properties of asymptotic cones of a finitely generated group and algorithmic properties of the group. I will show three methods for understanding the fundamental group of an asymptotic cone of a group and give examples of groups for each method.

** Wednesday, 29 February, 2012 **

Speaker: Darren Creutz (Vanderbilt University)

Title: The Property (T) "Half" of the Margulis-Zimmer Conjecture

Abstract: Generalizing the Margulis Normal Subgroup Theorem, Margulis and Zimmer
conjectured that any subgroup of a lattice in a higher-rank Lie group
which is commensurated by the lattice is (up to finite index) of a
standard form. I will present some of my work on property (T) for
totally disconnected groups and countable dense subgroups and explain
how it provides "half" of the solution to the conjecture. This is
joint work with Yehuda Shalom.

** Monday, March 12 - Friday, March 16, Workshop "Geometry and Analysis of Large Networks". **

** Wednesday, March 21, 2012 (joint with the Algebra seminar)**

Speaker: Chris Conidis (University of Waterloo)

Title: Proving that Artinian implies Noetherian without proving that Artinian implies finite length

Abstract: Let R be a commutative ring with identity. Recall from basic graduate algebra that:
1. R is Noetherian if it satisfies the ascending chain condition on its ideals;
2. R is Artinian if it satisfies the descending chain condition on its ideals; and
3. R is of finite length if there is a uniform bound on the length of any (strictly increasing/decreasing) chain of ideals in R.
It is well-known that 2. implies 1., but the proofs given in most standard algebra texts prove this by showing the stronger statement that 2. implies 3. This begs the question: "Can one prove that 2. implies 1. without showing that 2. implies 3? We will show that this is indeed the case by showing that, in the context of reverse mathematics, the former (weaker) statement is equivalent to weak K\"onig's lemma, while the latter (stronger) statement is equivalent to arithmetic comprehension in the context of omega-models. Another way to view our main result is that it constructs an omega-model of RCA (recursive comprehension axiom) in which 2. implies 1., but 2. does not imply 3.

** Wednesday, March 28, 2012 **

Speaker: Ashot Minasyan (University of Southampton, UK)

Title: Fixed point properties for group actions on simplicial and real trees

Abstract: I will discuss a construction of finitely generated groups that have non-trivial actions on real trees but which cannot act,
without fixing a vertex, on any simplicial tree. The talk will be based on a joint work with Martin Dunwoody.

** Wednesday, April 4, 2012 **

Speaker: Wes Camp (Vanderbilt University)

Title: Classification of right-angled Coxeter groups with locally connected CAT(0) boundary

Abstract: We provide conditions that determine exactly when a right-angled Coxeter group with no 3-flats has locally connected CAT(0) boundary, and discuss related results. We also provide examples to demonstrate why these results cannot be obtained from more standard techniques.

** Wednesday, April 11, 2012 **

Speaker: Michael Hull (Vanderbilt University)

Title: Quasi-cocycles on groups with hyperbolically embedded subgroups

Abstract: (Joint work with Denis Osin). We will present a general extension theorem which shows that a quasi-cocycle on a hyperbolically embedded subgroup can always be extended to the whole group. An application of this extension theorem shows that any group G containing a (proper and infinite) hyperbolically embedded subgroup has infinite dimensional second bounded cohomology with coefficents in l^{2}(G). Applying this theorem in another direction, we show that any hyperbolically embedded subgroup is undistorted with respect to stable commutator length, a result which appears to be new even for malnormal subgroups of free groups.