Topology & Geometric Group Theory Seminar, Spring 2013

Tuesdays and some Thursdays, 1:30 – 2:30, Malott 224

Tuesday, January 22, Anne Thomas (Sydney)

Divergence in right-angled Coxeter groups

The divergence of a pair of geodesic rays emanating from a point is a measure of how quickly they are moving away from each other. In Euclidean space divergence is linear, while in hyperbolic space divergence is exponential. Gersten used this idea to define a quasi-isometry invariant for groups, also called divergence, which has been investigated for classes of groups including fundamental groups of 3-manifolds, mapping class groups and right-angled Artin groups. I will discuss joint work with Pallavi Dani on divergence in right-angled Coxeter groups (RACGs). We characterise 2-dimensional RACGs with quadratic divergence, and prove that for every positive integer d, there is a RACG with divergence polynomial of degree d.

Tuesday, January 29, Benjamin Cooper (Universitat Zurich)

Some modular group actions on categories

There are a number of interesting braid group actions on categories. I will discuss these examples and introduce a way to construct categorical representations of the modular group from one of them.

Thursday, January 31, Jim Conant (University of Tennessee)

Topological interpretations of generalized Morita classes in the homology of Aut(F_n)

The definition of Morita classes in the homology of Aut(F_n) originally relied on an identification of the homology of Aut(F_n) with the stable cohomology of a certain complicated Lie algebra, and in particular an analysis of the abelianization of this Lie algebra. We show how these Morita classes can be reinterpreted in terms of homological gluing maps arising from gluing punctured 3-dimensional handlebodies together. In particular, statements about the abelianization of the Lie algebra can be translated into calculations of the homology of the mapping class groups (modulo torsion), Gamma_{n,s}, of punctured 3-dimensional handlebodies. In particular, a result of Conant-Kassabov-Vogtmann translates into a calculation of H_*(Gamma_{2,s}) in terms of modular forms and symmetric group representations. Known calculations in dihedral homology lead to a classification of $H_*(Gamma_{1,s})$. Not only does this point of view open up a whole new world of questions to explore, it gives a direct interpretation of how classes of small degree assemble to create classes of large degree. (Joint with Martin Kassabov and Karen Vogtmann)

Tuesday, February 5, Andrew Sale (Oxford and Cornell)

Finding Short Conjugators in Wreath Products and Free Solvable Groups

The question of estimating the length of short conjugators between elements in a group could be described as an effective version of the conjugacy problem. Given a finitely generated group $G$ with word metric $d$, one can ask whether there is a function $f$ such that two elements $u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free solvable groups, showing that f may be cubic. To do this we use the Magnus embedding, which allows us to see a free solvable group as a subgroup of a particular wreath product. This makes it helpful to understand conjugacy length in wreath products as well as metric properties of the Magnus embedding.

Tuesday, February 12, Johannes Cuno ( Graz University of Technology and Cornell)

The Tits alternative for non-spherical triangles of groups

Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalization of the amalgamated free product of two groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non-abelian free subgroup. We give another proof of this theorem. Moreover, we obtain two natural conditions, each of which ensures that the colimit of a euclidean triangle of groups either contains a non-abelian free subgroup or is virtually solvable. This result can be applied to extend a theorem by Kopteva and Williams about non-spherical Pride groups. (This is joint work with Jörg Lehnert.)

Tuesday, February 19 Matt Brin (Binghamton University)

Thompson's groups and the four color theorem

We describe the connection between Thompson's group F and the four color theorem and present an extension of F, some of whose elements seem to parametrize all the four colorings of planar maps.

Tuesday, February 26 Martin Kassabov (Cornell)

Groups of oscillating intermediate growth

(Joint work with I. Pak) We construct an uncountable family of finitely generated groups of intermediate growth, with growth functions of new type. These functions can have large oscillations between lower and upper bounds, both of which come from a wide class of functions. In particular, we can have growth oscillating between exp(n^a) (for 0.8 < a< 1) and any prescribed subexponential function, growing as rapidly as desired. Our construction is built on top of any of the Grigorchuk groups of intermediate growth, and is a variation on the limit of permutational wreath product.

Thursday, March 7, Ric Wade (University of Utah)

Centralizers of Dehn twists on free groups

There is a nice description of the centralizer of a Dehn twist in a mapping class group given by an exact sequence with the Dehn twist at one end and the mapping class group of a punctured surface at the other end (you obtain the punctured surface by cutting along the curve associated to the Dehn twist). We describe a similar picture for Dehn twists in Out(F_n). This is joint work with Moritz Rodenhausen.

Tuesday, March 12, Arthur Benjamin (Harvey Mudd) — joint Seminar with Combinatorics

Combinatorial trigonometry, alternating sums, and a method to DIE For

Many trigonometric identities, including the Pythagorean theorem, have combinatorial proofs. Furthermore, some combinatorial problems have trigonometric solutions. All of these problems can be reduced to alternating sums, and are attacked by a technique we call D.I.E. (Description, Involution, Exception). This technique offers new insights to identities involving binomial coefficients, Fibonacci numbers, derangements, and Chebyshev polynomials.

Tuesday, March 19, Spring Break

Thursday, March 28 Alessandra Iozzi (ETH Zurich) *****CANCELLED*****

The median class and superrigidity of actions on CAT(0) cube complexes

Let G be a group acting by automorphisms on a finite dimensional CAT(0) cube complex. We define the median class of the action in the second bounded cohomology of G with geometrically defined coefficients and prove that it does not vanish if the action is not elementary. We apply this result to establish a superrigidity result.

Thursday, April 4 Moira Chas (SUNY Stony Brook )

Normal distributions related to curves on surfaces

In an orientable surface with boundary, free homotopy classes of closed, oriented curves on surfaces are in one to one correspondence with cyclic reduced words in a minimal set of generators of the fundamental group. Given a cyclic reduced word, there are algorithms to compute the self-intersection of the corresponding free homotopy class (that is, the smallest number of self-crossings of a curve in the class, counted by multiplicity). With the help of the computer, one can make a histogram of how many free homotopy classes of twenty letters have self-intersection 0, 1, 2,.... The obtained histogram is essentially Gaussian. This experimental result led us to the following theorem, (jointly with Steve Lalley): If a free homotopy class of curves is chosen at random from among all classes of L letters, then for large L the distribution of the self-intersection number approaches a Gaussian distribution. The goal of this talk will be to discuss this theorem as well as related results and conjectures.

Tuesday, April 9 Brandon Seward (University of Michigan)

The Geometric Burnside's Problem

Burnside's Problem and the von Neumann Conjecture are classical problems from group theory which were long ago answered in the negative. In 1999, Kevin Whyte defined geometric analogs of these problems and proved the Geometric von Neumann Conjecture. In this talk I will present a proof of the Geometric Burnside's Problem. I will also present a strengthening of Whyte's result and draw conclusions about the existence of regular spanning trees of Cayley graphs.

Tuesday, April 16 Pallavi Dani, (Louisiana State University)

Filling invariants: homological vs. homotpical

Classical isoperimetric functions give optimal bounds on the minimal-area fillings of loops by disks. There are a number of variations: rather than sticking to loops and disks, one might consider fillings of spheres by balls, or cycles by chains. A natural question then is: for a given space, are these functions the same? Or are there spaces in which a particular type of filling (say filling cycles by chains) is more efficient than other types? What if the space admits a proper cocompact isometric action by a finitely presented group? I will talk about recent work with A. Abrams, N. Brady and R. Young which addresses this question.

Thursday, April 25 Milena Pabniak (IST Lisbon)

The Arnold Conjectures and an introduction to the generating functions technique

Symplectic diffeomorphisms posses certain rigidity properties (symplectic camel, non-squeezing theorem). An important manifestation of rigidity is given by the conjectures posed by V. Arnold describing a lower bound for the number of fixed points of a Hamiltonian diffeomorphism h (i.e. symplectomorphism homologous to identity) of a compact symplectic manifold, and a lower bound for the number of intersection points of a Lagrangian submanifold L and h(L). The conjectures have been proved in many special cases, but not in full generality. We will sketch the proof for C¹-small Hamiltonian diffeomorphisms. Then we introduce the technique of generating families which can be used to prove Arnold Conjectures for (some) other Hamiltonian diffeomorphisms. If time permits we will prove the Arnold Conjectures for complex projective spaces.

Tuesday, April 30 Federico Ardila, (San Francisco State)

The combinatorics of CAT(0) cubical complexes and robotic motion planning

A cubical complex is CAT(0) if "all its triangles are thin". These complexes play an important role in pure mathematics (group theory) and in applications (phylogenetics, robot motion planning, etc.). In particular, as Abrams and Ghrist observed, when one studies the possible states of a discrete robot, one often finds that they naturally form a CAT(0) cube complex. Gromov gave a remarkable topological/combinatorial characterization of CAT(0) cube complexes. We give an alternative, purely combinatorial description of them, allowing a number of applications. In particular, for many robots, we can use these tools to find the fastest way to move from one position to another one. The talk will describe joint work with Tia Baker, Megan Owen, Seth Sullivant, and Rika Yatchak. It will require no previous knowledge of the subject.