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
 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
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
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.