Topology & Group Theory Seminar

Vanderbilt University

Fall 2012/Spring 2013


Organizers: Michael Hull and Curt Kent

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

            Links to previous years seminar pages: 2010/2011 Schedule and 2011/2012 Schedule


Tuesday, 28 August 2012

            (At 1:10 pm in SC 1310)

o   Speaker: Gaven Martin (New Zealand Institute for Advanced Study)

o   Title: The solution to Siegel's Problem

o   Abstract: We outline the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram.   This solves (in three dimensions) the problem posed by Siegel in 1945.  (Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area).   There are strong connections with arithmetic hyperbolic geometry in the proof and the result has applications in the maximal symmetry groups of hyperbolic 3-manifolds (in much the same way that Hurwitz 84g-84 theorem and Siegel's result do).


 Wednesday, 12 September 2012

o   Speaker: Jesse Peterson

o   Title: Stabilizers of Ergodic Actions of Lattices and Commensurators, Part I

o   Abstract: A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well.

The proof of this theorem follows the same basic strategy as the proof of Margulis' Normal Subgroup Theorem. That is, one wants to show an object is finite by showing that it is amenable and has property (T).  In this context the object we are considering is the orbit equivalence relation generated by a non-free action of a commensurator.  A key step in this strategy is to first show that for a non-free action of a commensurator the orbit equivalence relation generated by the lattice is amenable.  Somewhat paradoxically, in order to show this one first has to consider contractive actions, introduced by Jaworski, which are extreme opposites of measure preserving actions. In the first talk I will discuss rigidity properties of these actions and explain how the key step described above follows from an intermediate factor theorem for these actions.


 Wednesday, 19 September 2012

o   Speaker: Darren Creutz

o   Title: Stabilizers of Ergodic Actions of Lattices and Commensurators, Part II

o   Abstract: A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Normal Subgroup Theorem for Commensurators of Lattices, due to the first author and Shalom, to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well.

In the second talk, I will explain how the results developed in the first talk lead to the conclusions about S-arithmetic lattices and to lattices in products.  The main ideas involve using the Howe-Moore property and property (T) to ensure that actions of the ambient groups satisfy the necessary conditions.  Another key idea in studying lattices in products is that most lattices in product groups are isomorphic to the commensurator of a lattice in one of the component groups.


 Wednesday, 26 September 2012

o   Speaker: Kate Juschenko

o   Title: On simple amenable groups I

o   Abstract: We will discuss amenability of the topological full group of a minimal Cantor system. Together with the results of H. Matui this provides examples of finitely generated simple amenable groups. Joint with N. Monod.


 Wednesday, 3 October 2012

o   Speaker: Kate Juschenko

o   Title: On simple amenable groups II

o   Abstract: We will discuss amenability of the topological full group of a minimal Cantor system. Together with the results of H. Matui this provides examples of finitely generated simple amenable groups. Joint with N. Monod.


 Wednesday, 10 October 2012

o   Speaker: John Ratcliffe

o   Title: On volumes of hyperbolic Coxeter polytopes and quadratic forms

o   Abstract: In this talk, the computation of the covolume of the group of units of the quadratic form x_1^2 + x_2^2 + \cdots + x_n^2 - dx_{n+1}^2 with $d$ an odd, square-free, positive integer, will be described. The covolume will be expressed in terms of Bernoulli numbers, Dirichlet $L$-functions, and powers of $\pi$.

John Mcleod has recently determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units for the case $d = 3$. We apply our covolume formula to determine the volumes of Mcleod's hyperbolic Coxeter polytopes. (a PDF of title and abstract)


 Wednesday, 17 October 2012

o   Speaker: Michael Brandenbursky

o   Title: Bi-invariant metrics on diffeomorphism groups.

o   Abstract: In this talk I will discuss various metrics on groups of diffeomorphisms of smooth manifolds, which do or do not preserve some additional structure (usually volume or symplectic form). Then I will restrict my talk to the case of the group G of area-preserving diffeomorphisms of the 2-disc. This group admits a natural bi-invariant Autonomous metric. I will show that any finitely generated free abelian group embeds bi-Lipschitz into the group G. This is essentially the same talk I gave at the Subfactor seminar.


 Wednesday, 31 October 2012

o   Speaker: Curt Kent

o   Title: Separating subsets in asymptotic cones

o   Abstract: I will talk about ends and separating subsets of asymptotic cones.  As well, I will show a construction of transversal R-trees in asymptotic cones with cut-points.


 Wednesday, 7 November 2012

o   Speaker: Nathan Habegger

o   Title: From quantum mechanics to the quantum computer, via topology.

o   Abstract: In 1987, the physicist Ed Witten gave a (physicist's) explanation showing that the Jones Polynomial of knots was in fact calculable from Chern-Simons Quantum Field Theory.  (For their separate contributions, both Ed and Vaughan were awarded the Fields Medal in 1990.)  Since these theories are topological in nature, in fact Jones/Witten had invented a whole new branch of Topology, called Quantum Topology, 25 years old today.

Quantum Topology studies more generally the notion of a TQFT (topological quantum field theory) and its perturbative analogues.  In the early 90's the author and his collaborators showed that the Jones Polynomial extended to invariants of knots in 3-manifolds, and that all of the axioms of a TQFT were satisfied.  (Perturbative aspects of these theories involve what has become to be known as the theory of Finite Type Invariants, first explored for knots by Vassiliev.  The Kontsevich Integral is the Universal such knot invariant.)

A TQFT can already be thought of as a Quantum Computer, since the Hilbert spaces involved are finite dimensional, at least if one defines a computer to be a finite set and operations thereupon, and defines a quantum computer to be the linear analogue (over the complex numbers).  If one wants to be a bit more restrictive, organizing a computer in terms of bits (a bit is a 2-pont set), then a quantum computer is just organized in terms of q-bits (a 2 dimensional complex Hilbert space or rather its projective analogue, the 2-sphere).  This is what one gets as the Hilbert space associated to a 2-sphere (not the one above) with 4 marked points, In theory then, 3-manifold/tangle pairs bounding such objects give (calculable) quantum computer operations.

Problems for the 21st century:
1. (Mathematics) Use these observations to do effective and interesting (useful) computations.
2. (Computer Science)  Implement these calculations on an (ordinary) computer.
3. (Condensed Matter Physics) Design circuits on an atomic scale which will do the same.


 Wednesday, 14 November 2012

o   Speaker: Scott Morrison

o   Title: Khovanov homology and 4-manifolds.

o   Abstract: I'll introduce Khovanov homology, a `categorical' extension of the Jones polynomial. Since the discovery of Khovanov homology, representation theorists have opened up a new world of categorical quantum groups. In this talk, I'll head in a different direction, explaining how the 4-dimensional nature of Khovanov homology makes it ideally suited for building a new 4-manifold invariant. I'll explain the construction, then discuss its present limitations and how we hope to get past them. (Joint work with Kevin Walker)


 Wednesday, 28 November 2012

o   Speaker: Alexander Ol'shanskii

o   Title: Bi-Lipschitz embeddings of groups

o   Abstract: This is a joint work with Denis Osin.  We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is.

We will briefly discuss some applications to subgroup distortion, compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F\o lner functions, and elementary classes of amenable groups.


Spring 2013


 Wednesday, 16 January 2013

o   Speaker: Remi Coulon (Vanderbilt University)

o   Title: Order and growth of automorphisms of free Burnside groups

o   Abstract: This is a work in progress with Arnaud Hilion.  The free Burnside group of rank r and exponent n B(r,n) is the largest group of rank r such the nth power of every element is trivial.  This object motivated many developments in group theory.  We are interested in its outer automorphism group. Every automorphism of the free group induces an automorphism of the Burnside group. We showed that its order as an automorphism of B(r,n) is related to its growth as an automorphism of the free group.  In this talk we will present the main steps of this proof and explain how this could be adapt to understand other properties of the automorphisms of B(r,n). 


 Wednesday, 23 January 2013

o   Speaker: Yuri Bahturin (Memorial University, Canada)

o   Title: Nilmanifolds and gradings on nilpotent Lie algebras

o   Abstract: We give a complete description of gradings by abelian groups on certain classes of nilpotent Lie algebras, including Lie algebras of maximal class. This has consequences concerning symmetries on homogeneous spaces of nilpotent Lie groups.

These results are joint with Elizabeth Remm and Michel Goze (Univesite de Haut Alsace, France).


 Friday, 1 February 2013 (in SC 1432)

o   Speaker: Kostya Medynets (US Naval Academy)

o   Title: Finite Factor Representations of Higman-Thompson Groups

o   Abstract: We will talk about relations between ergodic properties of group actions and the structure of group characters. The latter is equivalent to the classification of all finite-type factor representations. The outstanding conjecture (often attributed to Vershik) is that for a large class of groups their group characters must have the form \mu(Fix(g)), where $\mu$ is a G-invariant measure for some special group action on a measure space, Fix(g) is the set of all fixed points of group element $g$. We will then establish Vershik's conjecture for the family of Higman-Thompson groups. Since these groups have no non-trivial ergodic measures, we get that they have no non-trivial factor representations. Examples of other classes of groups known to satisfy Vershik's conjecture will be also discussed. The talk will be based on two recent preprints by Dudko and Medynets, "Finite factor representations of Higman-Thompson Groups" ArXiv 1212.1230 and "On characters of inductive limits of symmetric groups" Arxiv 1105.6325


 Wednesday, 20 February 2013

o   Speaker: Yves de Cornulier (Université Paris-Sud)

o   Title: On the compactommensurability and quasi-isometry classification of focal hyperbolic groups.

o   Abstract: The compactommensurability equivalence relation between locally compact groups is, by definition, generated by cocompact inclusions and quotients by compact subgroups. We give a full description of this equivalence relation in the context of focal hyperbolic locally compact groups, i.e. non-elementary amenable hyperbolic locally compact groups. A similar description holds for the quasi-isometry relation, but at a conjectural level. I'll state a conjectural general picture and describe partial results.


 Wednesday, 27 February 2013

o   Speaker: Jesse Peterson (Vanderbilt University)

o   Title: Character rigidity for special linear groups

o   Abstract: A character on a group is a positive definite function which takes the identity to 1 and is constant on conjugacy classes. Characters on a finite group gives an essential tool for understanding the representation theory of the group and motivated by this Thoma in 1964 initiated the study of characters on infinite groups. In 1966 Kirillov classified all characters on GL_n(k), and SL_n(k) for k a field and n \geq 2, excluding the case of SL_2(k). A number of other classification results have since been obtained for other groups by Ovcinnikov, Vershik, Kerov, and more recently by Bekka, Dudko, and Medynets, however the classification for SL_2(k) has not been completed. In my talk I will present the classification for SL_2(k) and SL_2 of some other rings and give some applications of these results. This is based on joint work with Andreas Thom.


 Wednesday, 6 March 2013

o   Spring Break


 


 Wednesday, 13 March 2013

o   Speaker: Denis Osin (Vanderbilt University)

o   Title: Acylindrically hyperbolic groups

o   Abstract: A group is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. This class encompasses many examples of interest: hyperbolic and relatively hyperbolic groups, Out(F_n) for n>1, all but finitely many mapping class groups, “most” fundamental groups of 3-manifolds, groups acting properly on proper CAT(0) spaces and containing rank 1 elements, 1-relator groups with at least 3 generators, etc. On the other hand, many non-trivial results known for hyperbolic groups can be generalized to acylindrically hyperbolic groups. The purpose of my talk is to survey some of the recent progress in this direction. I will also discuss some open problems.  The talk is partially based on my joint papers with F. Dahmani, V. Guirardel, M. Hull, and A. Minasyan


Wednesday, 20 March 2013

o   Speaker: Paul Schupp

o   Title: An asymptotic view of computability.

o   Abstract: After reviewing some results on genericity and generic computability in group theory, I will discuss the rich interaction of generic computability and the general theory of computation and then discuss coarse computability and coarse degrees.


 Wednesday, 27 March 2013

o   Speaker: Michael Mihalik (Vanderbilt University)

o   Title: Geodesically tracking quasi-geodesic paths for Coxeter groups.

o   Abstract: If (W,S) is a finitely generated Coxeter system we classify the quasi-geodesic paths (rays or lines) in the corresponding Cayley graph that are tracked by geodesics. The main corollary is that if W acts geometrically on a CAT(0) space X, then geometric geodesics in X are tracked by Cayley geodesics in X. This allows one to effectively transfer the group theory and combinatorics of (W,S) to help analyze the (local and asymptotic) geometry of X.


 Wednesday, 3 April 2013

o   Speaker: Michael Hull (Vanderbilt University)

o   Title: Small Cancellation in aclyindrically hyperbolic groups

o   Abstract: In a recent seminar, Osin introduced the class of aclyindrically hyperbolic groups and showed this class coincided with several previously studied and seemingly distinct classes of groups. In this talk, we will present a version of small cancellation theory for aclyindrically hyperbolic groups. We will discuss the main tools of this theory and some applications of these tools, especially the construction of various exotic quotient groups.


 Wednesday, 10 April 2013

o   Speaker: Alexander Dranishnikov (University of Florida)

o   Title: Macroscopic dimension and Gromov's conjecture

o   Abstract: In the 80’s, Gromov proposed a conjecture that the macroscopic dimension of the universal covering of a closed n-manifold with a positive scalar curvature metric does not exceed n-2. We prove his conjecture for manifolds with certain restrictions on the fundamental group. In particular, we prove Gromov's conjecture for exact virtual duality groups.


 Wednesday, 17 April 2013

o   Speaker: Wolfram Hojka (Vienna University of Technology)

o   Title: The harmonic archipelago from various viewpoints

o   Abstract: Different constructions are discussed for a space that has seen increased interest in wild algebraic topology -- the harmonic archipelago. Somewhat surprisingly, its fundamental group is independent of the choice of the underlying spaces. We discuss the homology as the abelianization, and show how another corresponding abelian group actually has greater discriminative power. Lastly, we investigate some mapping properties, such as mapping onto any separable profinite group.