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.

Mar. 20: Alex Taam, Graduate Center CUNY

posted Mar 15, 2015, 6:59 AM by NY Group Theory

 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.13, 16:00-17:00 A. Nikolaev (Stevens Institute)

posted Mar 7, 2015, 6:27 AM by NY Group Theory   [ updated Mar 7, 2015, 6:27 AM ]

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.

March 6, 16:00-17:00, A. Ushakov (Stevens)

posted Mar 1, 2015, 1:38 PM by NY Group Theory

Title: 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.

Feb 27, 16:00-17:00, Daniel Studenmund (Utah), Abstract commensurators of lattices in Lie groups

posted Feb 22, 2015, 8:11 PM by NY Group Theory

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.

Feb 20, 16:00-17:00, Room 5417. Doron Puder (Princeton)

posted Feb 17, 2015, 10:33 AM by NY Group Theory   [ updated Feb 17, 2015, 10:34 AM ]

Title: 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. 

Algebra Day CUNY Graduate Center, Dec 5, 9:30-5:00, Room 4102

posted Dec 3, 2014, 11:16 AM by NY Group Theory   [ updated Dec 3, 2014, 11:18 AM ]

Nov 21, Jennifer Taback, Bowdoin College

posted Nov 15, 2014, 1:49 PM by NY Group Theory

Quasi-isometry classification of the Baumslag-Gersten groups

I will present a quasi-isometry classification of the Baumslag-Gersten groups.  Gilbert Baumslag used a group in this family as an example of a non-cyclic one-relator group all of whose finite quotients are cyclic.  Gersten showed that the Dehn functions of groups in this family are bounded by towers of exponentials, and Platonov proved that each Dehn function is actually equal to a tower of exponentials.  I will describe how the geometric models of these groups are constructed from different Baumslag-Solitar complexes and how this geometry, combined with earlier rigidity results of Farb and Mosher for the solvable Baumslag-Solitar groups BS(1,n), yields a quasi-isometry classification for this family of groups which is perhaps more rigid than expected.  I'll include a brief overview of quasi-isometry classification and rigidity results for other families of groups whose geometric models are constructed from Baumslag-Solitar complexes. This is joint work with Tullia Dymarz. 

Nov 14, Yash Lodha, Cornell University, Room 5417

posted Nov 11, 2014, 9:50 AM by NY Group Theory

A new solution to the Van Neumann-Day Problem for finitely presented groups

Nov 7. Tim Riley Cornell University, (Room 5417)

posted Oct 29, 2014, 4:23 AM by NY Group Theory

 Taming the hydra: the word problem and extreme integer compression

 For a finitely presented group, the Word Problem asks for an algorithm which declares whether or not words on the generators represent the identity.  The Dehn function is the time-complexity of a direct attack on the Word Problem by applying the defining relations.

A "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions.  I will explain why, nevertheless, there are efficient (polynomial time) solutions to the Word Problems of these groups.  The main innovation is a means of computing efficiently with compressed forms of enormous integers.

 This is joint work with Will Dison and Eduard Einstein.

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