Topology & Group Theory Seminar

Vanderbilt University

2016/2017

Organizer: Mark Sapir

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

** Wednesday, August 31, 2016 **

Andrew Sale (Vanderbilt)

Title: When the outer automorphism groups of RAAGs are vast

Abstract:Right-angled Artin groups (RAAGs) are a class of groups that bridge the gap between free groups and free abelian groups. Thus, their outer automorphism groups give a way to build a bridge between GL(n,Z) and Out(F_{n}). We will investigate certain properties of these groups which could be described as a "vastness" property, and ask if it possible to build a boundary between those which are "vast" and those which are not.
One such property is as follows: given a group G, we say G has all finite groups involved if for each finite group H there is a finite index subgroup of G which admits a map onto H. From the subgroup congruence property, it is known that the groups GL(n,Z) do not have every finite group involved for n>2. Meanwhile, the representations of Out(F_{n}) given by Grunewald and Lubotzky imply that these groups do have all finite groups involved. We will describe conditions on the defining graph of a RAAG that are necessary and sufficient to determine when it's outer automorphism group has this property. The same criterion also holds for other properties, such as SQ-universality, or having a finite index subgroup with infinite dimensional second bounded cohomology.
This is joint work with V. Guirardel.

**Wednesdays, September 7, 14, 2016**

Denis Osin (Vanderbilt)

Title: Induced group actions on metric spaces.

Abstract: We will discuss the following natural extension problem for group actions: Given a group G, a subgroup H < G, and an action of H on a metric space S, when is it possible to extend it to an action of the whole group G on a possibly different metric space? When does such an extension preserve interesting properties of the original action of H? I will explain how to formalize this problem and will present a construction of the induced action of G which behaves well when G is hyperbolic relative to H or, more generally, H is hyperbolically embedded in G; in particular, the induced action solves the extension problem in these cases. This talk is based on a joint work with C. Abbott and D. Hume.

** Wednesday, September 21, 2016 **

Gili Golan (Vanderbilt)

Title: The generation problem in Thompson group *F*

Abstract: We show that the generation problem in Thompson group *F* is decidable, i.e., there is an algorithm which decides if a finite set of elements of *F* generates the whole *F*. The algorithm makes use of the Stallings 2-core of subgroups of *F*, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that *F* is a cyclic extension of a group *K* which has a maximal elementary-amenable subgroup *B*.

** Wednesday, October 5, 2016 **

John Ratcliffe (Vanderbilt)

Title: A Bieberbach theorem for crystallographic group extensions.

Abstract: Joint work with Steven Tschantz.
We will talk about our relative Bieberbach theorem: For each dimension *n*
there are only finitely many isomorphism classes of pairs of groups *(Γ,N)*
such that *Γ* is an *n*-dimensional crystallographic group and
*N* is a normal subgroup of *Γ* such that *Γ/N* is a crystallographic group.
This result is equivalent to the statement that for each dimension *n* there are only finitely many
affine equivalence classes of geometric orbifold fibrations of compact, connected, flat *n*-orbifolds.

** Wednesday, October 26, 2016 **

Mark Sapir (Vanderbilt)

Title: On planar maps of non-positive curvature

Abstract: This is a joint work with A. Olshanskii. We prove that if a (4,4)-map M does not contain regular (d x d)-squares, then the area of M does not exceed Cdn where C is a constant, and n is the perimeter of M. Similar properties are proved for (6,3) and (3,6)-maps. Thus if a van Kampen diagram over a small cancelation presentation does not contain large regular subdiagrams, then the area of the diagram is small.

** Wednesday, November 2, 2016 **

Stephen G. Simpson (Vanderbilt)

Title: Well partial orderings, with applications to algebra

Abstract:
A partial ordering consists of a set P and a binary relation < on P which is transitive (x < y < z implies x < z) and irreflexive (x is never < x). Within P, a descending chain is a sequence a > b > c > ..., and an antichain is a set of elements a, b, c, ... which are pairwise incomparable (neither a < b nor b < a). A well partial ordering is a partial ordering which has no infinite descending chain and no infinite antichain. To each well partial ordering P one can associate an ordinal number o(P). For example, the natural numbers N with their usual ordering form a well ordering of order type omega and hence a well partial ordering with o(N) = omega. The class of well partial orderings is closed under finite sums, finite products, and certain other finitary operations. As noted in a 1972 paper by J. B. Kruskal, well partial ordering theory is a "frequently discovered concept" with many applications, especially in abstract algebra (G. Higman, I. Kaplansky, ...). As a simple example, Dickson's Lemma says that for each positive integer k the finite product N^k is a well partial ordering, and this is the key to a proof of the Hilbert Basis Theorem: for any field K and positive integer k, the polynomial ring K[x_1,...,x_k] has no infinite ascending chain of ideals. The ordinal number involved here is omega^omega. There are also generalizations involving larger ordinal numbers such as omega^{omega^omega}. There is a subclass of the well partial orderings, the better partial orderings, which has stronger closure properties. For example, if P is a better partial ordering, then the downwardly closed subsets of P form a better partial ordering under the subset relation. This fact from better partial ordering theory can be used to prove that for any field K, the group ring K[S] of the infinite symmetric group S (the direct limit of the finite symmetric groups S_n as n goes to infinity) has no infinite ascending chain of two-sided ideals and no infinite antichain of two-sided ideals. There seems to be an open question as to how far this theorem can be generalized from S to other locally finite groups. R. Laver has used better partial ordering theory to prove that the countable linear orderings form a well partial ordering (or rather, a well quasi-ordering) under the embeddability relation. N. Robertson and P. Seymour have proved a difficult theorem: the finite graphs form a well quasi-ordering under the minor embeddability relation. I. Kriz has proved that the Friedman trees are well quasi-ordered under the gap embeddability relation.

** Wednesday, November 9, 2016 **

Ben Hayes (Vanderbilt)

Title: Metric approximations of wreath products

Abstract: I will discuss joint work with Andrew Sale. In it, we investigate metric approximations of wreath products. A mertic approximation of a group is a family of asymptotic homomorphisms into a class of groups so that the image of any nonidentity element is bounded away from zero. Metric approximations have received much recent interest and are related to several interesting conjectures, including Kaplansky's direct finiteness, Gottschalk's surjenctivity conjecture and the Connes embedding problem. Our results say the following: suppose that H is a sofic group. Then G wreath H is sofic (resp. linear sofic, resp. hyperlinear) if G is sofic (resp. linear sofic, resp. hyperlinear). No knowledge of sofic, linear sofic, or hyperlinear groups will be assumed.

** Wednesday, November 16, 2016 **

Vito Zenobi (Universit of Montpellier 2)

Title: The tangent groupoid and secondary invariants in K-theory.

Abstract: I will explain how to define secondary invariants that detect exotic structures on smooth manifolds or metrics with positive scalar curvatures on Spin Riemannian manifolds.
These invariants are elements in the K-theory of the tangent groupoid C*-algebra, introduced by Alain Connes to give a more conceptual viewpoint on index theory. These constructions
easily generalize to more involved geometrical situations (such as foliations), well encoded by Lie groupoids.

** Wednesday, November 30, 2016 **

Jing Tao (University of Oklahoma)

Title: Stable commutator lengths in right-angled Artin groups

Abstract: The commutator length of an element g in the commutator
subgroup [G,G] of a group G is the smallest k such that g is the
product of k commutators. When G is the fundamental group of a
topological space, then the commutator length of g is the smallest
genus of a surface bounding a homologically trivial loop that
represents g. Commutator lengths are notoriously difficult to compute
in practice. Therefore, one can ask for asymptotics. This leads to the
notion of stable commutator length(scl) which is the speed of growth
of the commutator length of powers of g. It is known that for n > 2,
SL(n,Z) is uniformly perfect; that is, every element is a product of a
bounded number of commutators, and hence scl is 0 on all elements. In
contrast, most elements in SL(2,Z) have positive scl. This is related
to the fact that SL(2,Z) acts naturally on a tree (its Bass-Serre
tree) and hence has lots of nontrivial quasimorphisms.
In this talk, I will discuss a result on the stable commutator lengths
in right-angled Artin groups. This is a broad family of groups that
includes free and free abelian groups. These groups are appealing to
work with because of their geometry; in particular, each right-angled
Artin group admits a natural action on a CAT(0) cube complex. Our main
result is an explicit uniform lower bound for scl of any nontrivial
element in any right-angled Artin group. This work is joint with Talia
Fernos and Max Forester.

** Wednesday, December 7, 2016 **

Gili Golan (Vanderbilt)

Title: Invariable generation of Thompson groups

Abstract: A subset S of a group G invariably generates G if for every choice of g(s) ∈ G,s ∈ S the set {s^{g(s)}:s ∈ S} is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariable generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

** Wednesday, December 14, 2016 **

Andrew Putman (University of Notre Dame)

Title: The high-dimensional cohomology of the moduli space of curves
with level structures

Abstract: We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least g-2. Well known conjectures of Looijenga would imply that this is sharp. This is joint work with Neil Fullarton.