Topology & Group Theory Seminar

Vanderbilt University

2016/2017

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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 of characteristic 0, 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 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.