# Spring 2015

Feb 6. 11:00-19:00 Joint meeting of NY Applied Algebra Colloquium , Model theory , Algebraic Cryptography and NY Group theory seminars, Alexei Fest.

http://web.stevens.edu/algebraic/AlexeiFest/

Feb 13: No seminar (before a long weekend)

Feb 20: Doron Puder, Princeton. **Free Groups and Measure Preservation**

** Abstract: **We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. This new characterization is related to structural phenomena in the set of f.g. subgroups of a given free group.

More specifically, for every finite group G, a word w in the free group on k generators induces a word map from G^k to G. We say that w is measure preserving with respect to G if given uniform distribution on G^k, the image of this word map distributes uniformly on G. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. In a joint work with O. Parzanchevski, we prove this conjecture.

Feb 27: Daniel Studenmund, University of Utah

**Abstract commensurators of lattices in Lie groups**

**Abstract: **The abstract commensurator of a group G is the group of all isomorphisms between finite index subgroups of G up to a natural equivalence relation. Commensurators of lattices in semisimple Lie groups are well understood using strong rigidity results of Mostow, Prasad, and Margulis. We will describe commensurators of lattices in solvable groups, where strong rigidity fails. If time permits, we will extend these results to lattices in certain groups that are neither solvable nor semisimple.

Mar 6: Sasha Ushakov, Stevens Institute

**Magnus embedding and algorithmic properties of groups F/N^(d)**

**Abstract.** Let N be a normal subgroup of a free group F. By N' we denote

the derived subgroup of N, and in general by N^(d) the dth derived subgroup.

In my talk I will discuss relations between the following algorithmic problems

for groups F/N^(d): word problem, power problem (membership into cyclic

subgroups), and conjugacy problem.

Joint work with Funda Gul and Mahmood Sohrabi.

Mar.13: Andrey Nikolaev, Stevens Institute

**Logspace and compressed word computation in finitely generated nilpotent groups.**

**Abstract:** Algorithmic problems in nilpotent groups have been extensively

studied, but few of the algorithms came with robust estimates of

computational complexity. In this talk we present some of the recent

results on the computation of normal forms, the membership problem,

the conjugacy problem, and computation of presentations for subgroups

in nilpotent groups. We show that these problems are solvable using

only logarithmic space and, simultaneously, in quasilinear time. We

also give polynomial time solutions to compressed-word versions of

these problems, in which each input word is provided as a

straight-line program. Time permitting, we will touch on other

applications of our techniques, such as finite separability questions,

distortion of embeddings, and solutions to other algorithmic problems.

This is a joint work with J.Macdonald, A.Myasnikov, S.Vassileva.

Mar. 20: Alex Taam, Graduate Center CUNY

**Effective JSJ decompositions of maximal $/Gamma$-limit quotients**

**Abstract:** Analogous to the decomposition of irreducible orientable closed 3-manifolds along incompressible tori from which the name is borrowed, a JSJ decomposition of a group can provide a canonical description of possible group splittings. I will define a JSJ decomposition for the class of finitely generated fully residually $\Gamma$ groups (i.e. $\Gamma$-limit groups), where $\Gamma$ is a fixed torsion-free hyperbolic group. Furthermore, given a group $G$ which is the coordinate group of a finite system of equations over $\Gamma$, I will show that there is an algorithm to find such decompositions of each maximal (with respect to a natural partial order) $\Gamma$-limit group which is a quotient of $G$. This is joint work with O. Kharlampovich and A. Myasnikov.

Mar 27: Nikolay Romanovskii, Novosibirsk State University

**Hilbert's Nullstellensatz in algebraic geometry over rigid solvable groups**

**Abstract: **The classical Hilbert's Nullstellensatz says: if $K$ is algebraically closed field and there is a system of polynomial equations over $K$, $\{f_i (x_1, \ldots, x_n) = 0 \ | \ i \in I \}$, then an equation $ f (x_1, \ldots, x_n) = 0 $ is a logical consequence of this system (satisfies all the solutions of the system in $K^n$) if and only if some nonzero power of $f$ belongs to the ideal $ (f_i \ | \ i \in I) $ of the ring $ K [x_1, \ldots, x_n]. $ One can say say that we give an algebraic method for constructing all logical consequences of the given system of equations: $f$ is obtained from $f_i \ (i \in I)$ using the operations of addition, subtraction, multiplication by elements of $K [x_1, \ldots, x_n]$, and extraction of roots.

Our approach to Hilbert's theorem in algebraic geometry over groups is as follows.

1. We should consider some good class of equationally Noetherian groups, let it be a hypothetical class $\mathcal{K}$.

2. In this class, we need to define and allocate an algebraically closed objects and to prove that any group of $\mathcal {K}$ is embedded into some algebraically closed group. Hilbert's theorem should be formulated and proved for algebraically closed in $\mathcal{K}$ groups.

3. Further, let $G$ be an algebraically closed group in $\mathcal{K}$. We think about equations over $G$ as about expressions $v=1$, where $v$ is an element of the coordinate group of the affine space $G^n$.

4. Since an arbitrary closed subset of $G^n$ is defined in general not by a system of equations, but by a positive quantifier free formula (Boolean combination without negations of a finite set of equations) we should consider as basic blocks not equations, but positive formulas.

5. We should specify and fix some set of algebraic rules of deduction on the set of positive formulas over $G$.

6. If the above conditions Hilbert's theorem will consist in a statement that all logical consequences of given positive formula over $G$ are exactly the algebraic consequences.

We realized this approach in algebraic geometry over rigid solvable groups.

Apr.3: no seminar (spring break)

Apr.10: no seminar (spring break)

**Apr.17: Bruce Kleiner, NYU, "Boundaries of hyperbolic groups"**

**Abstract**: Every hyperbolic group has a boundary at infinity, which is a compact

space on which the group acts by homeomorphisms. The boundary carries additional canonical structure in the form of a family of (visual) metrics; in the case of lattices in Lie groups, this metric structure has played a key role in many results, including Mostow's rigidity theorem.

After reviewing the background, the lecture will discuss some further applications of this structure, and related open problems.

Apr.24: Conference "Infinite Group Theory: From the Past to the Future" A conference celebrating the great achievements

in infinite group theory over the past two decades and in honor of the 70th birthdays of G. Rosenberger and D. Spellman

http://faculty.fairfield.edu/pbaginski/InfiniteGroupTheory2015.html

AFTERNOON SESSION 2 – ROOM 5417 4:15-5:00 – Eugene Plotkin (Bar Ilan) “Equations over algebras: The logical geometry”

**May 1: Lee Mosher (Rutgers) Title: Hyperbolic actions and second bounded cohomology for subgroups of**

**Out(F_n) (joint with Michael Handel)**

Abstract: After surveying the co-evolution of the theories of hyperbolic actions and of second bounded cohomology of groups, we will report on recent progress on this topic for subgroups of Out(F_n).

May 8: Andy Putman (Rice U.) **TITLE: The stable cohomology of congruence subgroups**

**ABSTRACT:**

I will explain how to use representation-theoretic tools to understand patterns in the cohomology of congruence subgroups of SL(n,Z) and related groups. This is joint work with Steven Sam.