# Spring 2016

Feb.5 B. Farb (U. of Chicago)

Title: Representation theory and the homology of finite covers of graphs and surfaces.

Abstract: In this talk I will explain some problems and progress on basic questions about

the homology of finite index subgroups of free groups and surface groups. The story will

begin with a beautiful theorem of Chevalley and Weil from 1934. This work is joint

with Sebastian Hensel.

Feb.12 - no seminar

Feb.19 O. Kharlampovich (Hunter College CUNY)

Title: Equations and elementary classification questions in a group algebra of a free group.

Abstract: We will show that equations are undecidable in

a group algebra of a free group (or torsion free hyperbolic group) over a field of characteristic zero. This is a surprising result that shows that answers to first-order questions in a free non-abelian group and in its group algebra are completely different. (Joint results with A. Miasnikov)

Feb.26 K. Bou-Rabee (CCNY)

Title: The topology of local commensurability graphs

Abstract: Let $G$ be a group and $p$ a prime number. The $p$-local commensurability graph of $G$ has all finite-index subgroups of $G$ as vertices, and edges are drawn between $A$ and $B$ if $[A : A\cap B][B : A\cap B]$ is a power of $p$. In this talk, I will talk on recent work that draws group-theoretic properties from the diameters of components of these graphs. The connected diameter of a graph is the supremum over all graph diameters of every component of the graph. Every $p$-local commensurability graph of any nilpotent group has connected diameter equal to one, and the reverse implication is true. In contrast, every nonabelian free group has connected diameter equal to infinity for all of their $p$-local commensurability graphs. Moreover, solvable groups, in a sense, fill in the space between free groups and nilpotent groups. This talk covers joint work with Daniel Studenmund and Chen Shi.

Mar.4 M. Sohrabi (Stevens) Title: On groups elementarily equivalent to *T(n,R)*

Abstract: In this talk I will present my joint results with Alexei Miasnikov on the structure of a group elementarily equivalent to the solvable group *T(n,R) *of all invertible upper triangular *nxn* matrices over a characteristic zero integral domain where* n *is greater than or equal to 3. In particular, I will talk about necessary and sufficient conditions for a group being elementarily equivalent to *T(n,R)* where *R* is a characteristic zero algebraically closed field, a real closed field, a number field, or the ring of integers of a number field. Time permitting I'll also present a new way to look at Belegradek's work on elementary theory of the group *UT(n,R) *of upper unitriangular matrices via lattice groups and Lie algebras.

Mar.11 I. Lysenok (Stevens) Quadratic equations in free metabelian groups.

Mar.18 J. Maher (CUNY) Title: Random walks on weakly hyperbolic groups

Abstract: Let G be a group acting by isometries on a Gromov hyperbolic

space, which need not be proper. If G contains two hyperbolic elements

with disjoint fixed points, then we show that a random walk on G converges

to the boundary almost surely. This gives a unified approach to

convergence for the mapping class groups of surfaces, Out(F_n) and

acylindrical groups. This is joint work with Giulio Tiozzo.

Mar.25 - no seminar

Apr.1 A. Myropolska ( University of Paris-Sud)

Title: Nielsen and Andrews-Curtis equivalence in finitely generated groups

Abstract: Various aspects of geometric group theory lead to the study of the natural

action of Aut(F_n) on the set Epi(F_n, G) of generating n-tuples of a group

G generated by at least n elements. One of the main questions, raised in

the context, is the transitivity of this actions.

In the talk, we will give an introduction to the subject, then extensively

discuss its relation to the Andrews-Curtis conjecture and explain the

transitivity results for the class \emph{MN} of finitely generated groups

of which every maximal subgroup is normal (this includes nilpotent groups

and Grigorchuk-like groups).

Apr.8 M. Hagen (U. of Cambridge)

Title: Curve complexes for cube complexes

Abstract: I'll discuss a hyperbolic space -- the "contact graph" --

associated to a CAT(0) cube complex. Using the example of a

right-angled Artin group, I'll illustrate how the contact graph can help

one understand the large-scale geometry of a CAT(0) cube complex in very

much the same way that the curve graph of a surface can, by work of

Masu-Minsky, be used to understand the geometry of the mapping class

group. This is joint work with Jason Behrstock and Alessandro Sisto.

Apr.15 M. Volkov

Title: Algebraic properties of monoids of diagrams and 2-cobordisms.

Abstract: Partition of diagram monoids first appeared in 1937

in a paper by Brauer in which they serve as vector space bases

of certain associative algebras relevant in representation theory

of classical groups. Other species of diagram monoids were

invented by Temperley and Lieb in the context of statistical

mechanics in the 1970s and by Kauffman and Jones in the context

of knot theory in the 1980s. Since then diagram monoids have

revealed many other connections, e.g., with low-dimensional

topology, topological quantum field theory, quantum groups etc.

Recently, they have been intensively studied as purely algebraic

objects, and these studies have shown that diagram monoids are

quite interesting from this viewpoint as well.

In the talk, we first present geometric definitions for some

classes of infinite diagram monoids and then survey our results

on the finite basis problem for their identities. Whilst it is

not clear whether or not a study of the identities of infinite

diagram monoids may be of any use for any of their non-algebraic

applications, such a study has constituted an interesting challenge

from the algebraic viewpoint and required to develop new techniques.

We also report on a recent application of these new techniques to

the finite basis problem for the identities of monoids of 2-cobordisms.

Apr.22 - Spring Recess

Apr.29 L. Babai (U. of Chicago) Room 4102, Science Center

Title: A little group theory goes a long way: The group theory

behind recent progress on the Graph Isomorphism problem

Abstract:

One of the fundamental computational problems in the complexity

class NP on Karp's 1973 list, the Graph Isomorphism problem (GI)

asks to decide whether or not two given graphs are isomorphic.

While program packages exist that solve this problem remarkably

efficiently in practice (McKay, Piperno, and others), for

complexity theorists the problem has been notorious for its

unresolved asymptotic worst-case complexity: strong theoretical

evidence suggests that the problem should not be NP-complete, yet

the worst-case complexity has stood at $\exp(O(\sqrt{v\log v}))$

(E. M. Luks, 1983) for decades, where $v$ is the number of

vertices.

By addressing the bottleneck situation for Luks's algorithm, we

recently reduced this ``moderately exponential'' upper bound to

quasipolynomial, i.e., $\exp((\log v)^c)$.

The problem we actually solve in quasipolynomial time is more

general: we solve the String Isomorphism problem (SI), introduced

by Luks in his seminal 1980/82 paper in which he brough in-depth

applications of group theory to bear on the GI and SI problems.

The input to an instance of SI is a permutation group $G$ acting

on a set $\Omega$ of $n$ elements, and a pair of strings, $x$ and

$y$, over $\Omega$ (functions that map $\Omega$ to a finite

alphabet). The question is, does there exist a permutation in $G$

that transforms $x$ into $y$ (``anagrams under group action'').

($G$ is given by a list of generators.) As Luks pointed out, this

problem is polynomial-time equivalent to the Coset Intersection

problem: given two subcosets of the symmetric group $S_n$, decide

whether or not their intersection is empty.

The following group theoretic lemma is at the heart of the new

algorithm.

Let $G$ be a permutation group of degree $n$ and $f$ an

epimorphism of $G$ onto $A_k$, the alternating group of degree

$k$. We say that a point $p$ in the permutation domain on which

$G$ acts is _affected_ by $f$ if the stabilizer $G_p$ is mapped by

$f$ to a proper subgroup of $A_k$. Let $U$ be the set of

unaffected points and let $H$ be the pointwise stabilizer of $U$

in $G.$

Unaffected Stabilizers Lemma.

If $k > \max\{8, 2+log_2 n\}$ then $f$ maps $H$ onto $A_k.$

In the talk we outline the proof of this result and try to convey

the basic idea, how, through the Lemma, the affected/unaffected

dichotomy plays a central role in the design and analysis of

the algorithm.

The proof of the lemma is elementary with reference to Schreier's

Hypothesis that the outer automorphism group of every finite

simple group is solvable. Schreier's Hypothesis follows from the

Classification of Finite Simple Groups (CFSG). Under the slightly

stronger assumption that $k > (log n)^c$ for some constant $c$,

Laszlo Pyber recently announced a CFSG-free proof of the result.

The paper is available at arXiv:1512.03547.

Helpful reading:

E.M. Luks : Isomorphism of graphs of bounded valence can be

tested in polynomial time. J. Comp. Sys. Sci., 25:42--65, 1982.

May 6 M. Bestvina (Utah)

Title: On the Farrell-Jones conjecture for mapping class groups

Abstract: I will try to describe what the Farrell-Jones conjecture is

about, and how one goes about proving it. Then I will try to outline a

proof of FJC for mapping class groups, which is work in progress,

joint with Arthur Bartels.

May 13 E. Zelmanov (U. California, San-Diego), Room 4102

Groups with Identities.

Abstract. We will discuss groups satisfying pro-p and prounipotent identities : examples, theory and possible applications.