The Hebrew University of Jerusalem

## Amitsur Algebra Seminar

#### Meeting time and place: Thursday 12:00-13:15 in Manchester 209

Schedule for Semester B: February 2010 - June 2010

 Date Speaker Title Abstract May 20, 2010Alex Lubotzky (HUJI)On Rank gradient, Heegard splitting and Cost We will present a paper of M. Abert and N. Nikolov. In that prepring they consider two conjectures- one in 3-dim. geometry (saying that the number of generators of the fundamental group of an hyperbolic 3-manifold is equal to its Heegard genus) and one in measurable group theory ( "Fixed cost conjecture"). There main result is that (at least) one of these (seemingly unrealetd!) conjectures is false! They prove it by relating the cost of the action of a group on its profinite completion to its rank gradient. All the relevant notions will be explained. May 6, 2010Boaz Tsaban (BIU)The Algebraic Eraser and short expressions of permutations as products On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the Algebraic Eraser scheme for key agreement over an insecure channel, using a novel hybrid of infinite and finite noncommutative groups. They also introduced the Colored Burau Key Agreement Protocol (CBKAP), a concrete realization of this scheme.We present general, efficient heuristic algorithms, which extract the shared  key out of the public information provided by CBKAP. These algorithms are, according to heuristic reasoning and according to massive experiments, successful for all sizes of the security parameters, assuming that the keys are chosen with standard distributions.Our methods come from probabilistic group theory, and have not been used before in cryptanalysis. In particular, we provide a(n embarrassingly) simple and efficient heuristic algorithm for finding short expressions of permutations in S_n, as products of given random permutations.Our algorithm gives (heuristically and experimentally) expressions of length O(n^2\log n), in time O(n^4\log n) and space O(n^2\log n), and is the first practical one for n>128.The talk is self-contained and was already delivered successfully to students. So, students are welcome.Joint work with Arkadius Kalka and Mina Teicher (BIU).Disclaimer: Our work does not make any claim concerning the security of the actual distributions used in the Algebraic Eraser (TM). These distributions are are proprietary and are not available to the authors April 29, 2010 Amnon Yekutieli (BGU) Central Extensions of Gerbes The talk will begin with an explanation of the concept of extension of groupoids. This is a mild generalization of the familiar concept of extension of groups. Oddly this concept appears to be new; perhaps because it is not very interesting... Next I will explain what is a gerbe. This is a very complicated concept. My point of view is that a gerbe G, on a topological space X, is the geometric version of a nonempty connected groupoid -- much in the same way that a sheaf of groups on X is the geometric version of a group. I will try to avoid the horrid technicalities of 2-categories and stacks. The structure of nonabelian gerbes is known to be very complicated. Mainly one wants to know if a given gerbe G is trivial. The theory of nonabelian cohomology (Giraud, 1960's) was invented for this purpose. However this theory is too abstract to be useful. An extension of gerbes is gotten by geometrizing the concept of extension of groupoids. Now this is a very interesting concept. A special case is that of central extension of gerbes, and I will give some examples. Next I will state a couple of results about obstruction theory for central extensions of gerbes. This theory gives concrete criteria to determine if a given gerbe is trivial. A more elaborate result is about the structure of pronilpotent gerbes. If time permits I will explain how the last result is used in my work on twisted deformation quantization of algebraic varieties. The paper (same title as lecture) is available online as arXiv:0801.0083; to appear in Advances Math. April 22, 2010Amitai Regev (Weizmann)A characterization of standard polynomials by codimension growth Given a p.i. algebra A, it has a sequence of codimentions c_n(A). A deep theorem of Giambruno and Zaicev says that as n goes to infinity, the limit of the n-th root of c_n(A) always exist (and is an integer). That integer, denoted exp(A), is called the exponent of the algebra A. Given a polynomial f, one considers U(f), the relatively free p.i. algebra satisfying f=0, then denotes exp(f):= exp(U(f)). Given n\ge 6, we show that if f=St_n is the standard polynomial of degree n, and g is any polynomial of degree n which is not a multiple of St_n, then exp(g) < exp(St_n). March 25 - April 9: Pesach vacation March 18, 2010David El-Chai Ben-Ezra (HUJI) O’Nan Scott Theorem The above is a classification theorem of the primitive finite groups. Primitive groups in certain meanings are the “atoms” of permutation groups, and, therefore, have great importance in the study of their structure. Moreover, the point stabilizers of a primitive group form a conjugacy class of maximal subgroups, so classification of primitive groups is closely related to the study of maximal subgroups. It turns out that the key to analyzing finite primitive groups is to study the socle, which is the subgroup generated by the minimal normal subgroups.In the lecture we will study the structure of the socle of primitive finite groups, and will give a brief description of the types of the primitive finite groups.The lecture will be given in Hebrew, and only basic knowledge of group theory is needed. March 11, 2010Uri Shapira (HUJI)Applying dynamics to Number Theory I will describe a recent joint work with Manfred Einsiedler and Lior Fishman in which we use rigidity results in dynamics to solve several problems regarding continued fraction expansions. In particular, we settle the following conjecture of M. Boshernitzan: For a real number x, let a_n(x) be the n'th coefficient in the continued fraction expansion of x and let c(x) = limsup a_n(x)  (the number c(x) has a clear geometric meaning that I will explain in the talk). Theorem (EFS): For any irrational x, the sequence c(nx) is unbounded.The talk is intended for general audience and will not assume prior knowledge in the subject. March 4, 2010Chen Meiri (HUJI)The congruence subgroup property for Aut(F_2) Asada proved that Aut(F_2) has the congruence subgroup property. His proof was based on techniques involving Galois extensions of rational function fields of algebraic curves. In this talk we will present a group-theoretic proof of Bux, Ershov and Rapinchuk for this theorem. February 25, 2010Doron Shafrir (HUJI)Duality of orbits and invariant polynomials in algebraic actions Given a group G acting on a finite dimensional vector space V, it is natural to try to parametrize the orbits of the action. This is the goal of invariant theory. For example, if the group is finite the orbits are parametrized by point of an affine variety. When the group and action are algebraic it is possible to give a variety that corresponds to "most" (Zariski open subset) orbits. To find the quotient variety it is important to understand the ring of invariants k[V]^G. In this talk I will discuss properties of this ring in connection with the orbits. For example, we will see that when the ring is polynomial, and if G has no characters, then every orbit of maximal dimension is a component of a set of invariant equations.The talk is part of the Action Now day, and therefore will be held in Canada Elion and will start at 12:10.

Schedule for Semester A: October 2009 - January 2010

Schedule for Semester B: February 2010 - June 2010

#### Schedule for Semester A: October 2009 - January 2010

Date

Speaker

Title

Abstract
Jan 21, 2010

Leonid Fel (Technion)

New Identities for Degrees of Syzygies in Numerical Semigroups
We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series $H\left({\bf d}^m;z\right)$ of nonsymmetric numerical semigroups ${\sf S}\left({\bf d}^m\right)$ of arbitrary generating set of positive integers ${\bf d}^m=\left\{d_1,\ldots,d_m\right\}$, $m\geq 3$.

These identities were obtained by studying together the rational representation of the Hilbert series $H\left({\bf d}^m;z\right)$ and the quasipolynomial representation of the Sylvester waves in the restricted partition function $W\left(s,{\bf d}^m\right)$. In the cases of symmetric semigroups and complete intersections these identities become more compact.
Jan 14, 2010

Uri Shapira (HUJI)

A neat proof of the Borel Harish-Chandra theorem
The talk is aimed towards students. I will describe a beautiful argument (of Margulis if I'm not mistaking), to prove the above theorem.  Along the way I will give the necessary introduction to Mautner phenomena, nondivergence of unipotents and algebraic groups. It shouldn't matter if the audience doesn't know anything about the above.
Jan 7, 2010

Sebastian Petersen (UNIBW)

Monodromy of abelian varieties over finitely generated fields
(Joint work with Wojciech Gajda)
Let $A$ be an abelian variety over a finitely generated field $K$. For every prime number $l\neq char(K)$ we let $K_l=K(A[l])$ be the field obtained by adjoining (the coordinates of) all the $l$-torsion points of $A$ to $K$. Then $K_l/K$ is a Galois extension and the Galois group $G(K_l/K)$ is known as the mod-$l$ monodromy group of $A/K$. Chris Hall determined these monodromy groups for a certain class of abelian varieties over a global field $K$. We explain Hall's result and a generalization due to ours. We use this to obtain a partial result towards the following conjecture of Geyer and Jarden.

Conjecture: Let $A$ be a non-zero abelian variety over a finitely generated field $K$.

a) For almost all $\sigma\in G_K$ there are infinitely many prime numbers $l$ such that the group $A(K_s(\sigma))[l]\neq 0$.

b) Let $e\ge 2$. For almost all $\sigma\in G_K^e$ the group $A(K_s(\sigma))_{TORS}$ is finite.

Here $K_s(\sigma)$ stands for the fixed field in the separable closure $K_s$ of the subgroup generated by the components of the vector $\sigma\in G_K^e$. "Almost all'' is meant in the sense of Haar measure.
Dec 31, 2009

Lior Bary-Soroker (HUJI)

Haran's diamond theorems
In 1999 Dan Haran established two diamond theorems', the first giving a permanence criterion for Hilbertian fields [1] and the second for for free profinite groups of infinite rank [2]. Recently there have been several generalizations, namely diamond theorem' for finitely generated free profinite groups (BS), for semi-free profinite groups (BS-Haran-Harbater), fully Hilbertian fields (BS-Paran), and surface groups (BS-Stevenson-Zalleskii). Also these diamond theorems were the half way in Jarden's construction of fields with free absolute Galois groups.

In this talk we shall discuss Haran's original theorem (probably in the group theoretic setting) and try to provide some insights of its proof.

[1] D. Haran, Hilbertian Fields under Separable Algebraic Extensions, Inventiones mathematicae 137, 85–112 (1999).
[2] D. Haran, Free subgroups of free profinite groups, Journal of Group Theory 2, 307–317 (1999).

Lior's blog
Dec 24, 2009

Hilbertianity of fields of power series
Let $R[[X]]$ be the ring of formal power series over a Noetherian domain $R$, and let $F$ be its quotient field. We prove that $F$ is Hilbertian. More generally, this holds for any domain $R$ contained in a rank-1 valuation ring of its quotient field. This result gives a positive solution to an open problem of Jarden. As a corollary, we strengthen previous Galois theoretic results over such fields, obtained by Lefcourt, Harbater-Stevenson, Pop, and the speaker.

The talk will include a background to these notions - no prior knowledge is needed.
Dec 17, 2009

Amichai Eisenmann (HUJI)

Counting Arithmetic Subgroups in PSL_2 over a p-adic field
Let H denote PSL_2 over some local field, endowed with a fixed Haar measure. Let AL_H(x) denote the number of conjugacy classes of arithmetic lattices of co-volume <=x. It has been recently shown by Belolipetsky, Gelander, Lubotzky and Shalev that for H=PSL_2(\mathbb{R}) one has the following precise result: lim \frac{log AL_H(x)}{x logx}=\frac{1}{2\pi} (The Haar measure is the one defined by Riemannian measure on hyperbolic plane)

I intend to describe how one can prove a theorem of similar nature for PSL_2 over a p-adic field, for a large family of p-adic fields. The proof uses ideas from the proof of the theorem above, as well as results of Schlage Puchta described in the previous talk.
Dec 10, 2009

Amichai Eisenmann (HUJI)

Subgroup growth of virtually free groups
Let \Gamma be a group. Denote by s_n(\Gamma) the number of subgroups of \Gamma of index at most n. The growth coefficient of \Gamma is defined by  \nu(\Gamma):=limsup \frac{log s_n(\Gamma)}{n logn}.

Recently it has been shown by J.C. Schlge-Puchta that if \Gamma is a fundamental group of a finite graph of finite groups (which is equivalent to being finitely generated and virtually free) then \nu(\Gamma) is the solution of a linear optimization problem with rational coefficients, which can be effectively computed from the graph of groups.
I intend to describe the proof to this theorem.

Next I would like to show how this theorem can be used in order to prove that
\nu(\Gamma)=-\chi(Gamma)  (where \chi(\Gamma) denotes the Euler-Poincare char.)
whenever \Gamma is a lattice in PSL_2 over a p-adic field K , for a large family of p-adic fields K (in particular Q_p for p>3).
If time permits I hope to indicate the connection with the problem of counting lattices in PSL_2 over such fields.

The elements of the theory of graph of groups necessary for the talk will be briefly sketched.
No prior knowledge of linear optimization theory is supposed.
Dec 3, 2009

Menny Aka (HUJI)

Profinite completions and Kazhdan's property (T)
The profinite completion $\widehat \Gamma$ of a finitely generated residually-finite group $\Gamma$, contains a lot of information on $\Gamma$; as $\widehat \Gamma$ has more structure than $\Gamma$, much effort is concentrated on understanding to what extent the properties and structure of $\Gamma$ are determined by $\widehat \Gamma$.
A property $\cP$ of finitely generated residually finite groups is called a \emph{profinite property} if the following is satisfied: If $\Gamma_1$ and $\Gamma_2$ are such groups with $\widehat{\Gamma_1} \cong \widehat{\Gamma_2}$ (i.e. the profinite completion of $\Gamma_1$ and $\Gamma_2$ are isomorphic) then $\Gamma_1$ has $\cP$ if and only if $\Gamma_2$ has $\cP$.

In the talk I will briefly survey some interesting properties which are profinite and some which are not. In a work in preparation, Kassabov showed that property $(\tau)$ is not a profinite property, and he asked whether Kazhdan property (T) is a profinite property. In the main part of the talk, I will prove that Kazhdan's property (T) is not a profinite property. I will present two arithmetic groups which have isomorphic profinite completion, where one has property (T) and the other does not.

This result is a byproduct of a general question of determining how many isomorphism classes of arithmetic groups can have an isomorphic profinite completion. If time allows, I'll show  constructions of many non-trivial examples of non-isomorphic arithmetic groups with isomorphic profinite completions as the one above, and sketch a proof of the following result -  every set of arithmetic groups with isomorphic profinite completion consists of \emph{finitely many} isomorphism classes.

Nov 26, 2009

Seymour Bachmuth (UCSB)

Solution of Burnside’s Problem

Burnside’s Problem asks whether a finitely generated group is finite if the orders of its elements are bounded. We give a solution for groups of prime power exponent, but restrict our presentation to 2-generator groups. The proof is surprisingly elementary and should be completely understandable to anyone with a basic knowledge of groups and rings.

Nov 19, 2009

Alexei Kanel

Construction of nill- but not nilpotent semigroup

The talk is devoted to the construction of finitely presented infinite  nill-semigroup. This problem was posed by L.Shevrin and M.Sapir. Joint work  with Iliya Ivanov.

Such semigroup can't be constructed via monomial relations: if there is finite set of forbiden words and infinite word (superword) without forbiden subwords, then there exist also a periodic uperword without forbidden subwords. From other hand there exist a finite set of prototipes and contacting rules on the plane such that only non-periodic tilling exist.

According these tillings one can construct a rewriting rule system. In order to construct infinite nill semigroup one needs specific metric space. The construction is based on the construction of metric space such that any two points on the distance $R$ can be connected with family of geodesics covering a disc of width  $\lambda R$ and aperiodic tiling on that space.

Nov 12, 2009

Konstantin Golubev (HUJI)

Cayley graphs and dessins d’enfants

Dessins d'enfants (maps on surfaces) are parameterized by conjugacy classes of subgroups of Grothendieck oriented cartographic group. Modular group PSL_{2}(Z) is isomorphic to the factor-group of the cartographic group. Finite index subgroups of modular group parameterize regular 3-valent dessins d'enfants.

We introduce an "extended" modular group {EPSL_{2}(Z)  (it contains PSL_{2}(Z) as a subgroup of index 2) and a universal dessin d'enfant. With the help of these two objects we prove the following theorem: Cayley graph of a factor-group of the extended modular group by a subgroup of finite index is dual to a quotient of the universal dessin d'enfant by the same subgroup.

Nov 05, 2009

Andy Magid (Oklahoma)

The Complete Differential Galois Closure of a Differential Field

A differential field F is a field with a derivation D; that is, an additive endomorphism satisfying D(ab)=aD(b)+bD(a). The constant field C of F is the kernel of D. C is assumed algeraically closed and characteristic zero. Let L=D^n+a_1D^{n-1}+ \dots a_nD^0 be a (linear, monic) differential operator over F. The (unique) field generated over F by a full set of solutions to the (homogeneous) differential equation L=0, and containing no new constants, is called the differential Galois, or Picard--Vessiot, extension of F for L. The compositum of all of these for all L is the Picard--Vessiot closure of F. Such closures may have proper differential Galois extensions, hence closures of their own. Taking these and iterating gives the complete Picard--Vessiot closure. We discuss what is known about this field, its group of differential authomorphisms, and the correspondence between the latter's subgroups and differential subfields of the complete closure.

Oct 29, 2009

Avinoam Mann (HUJI)

The growth of free products

It is known that free products (with or without amalgamations), HNN extensions, 1-related groups, and groups of positive deficiency, usually have uniformly exponential growth, with explicitly given lower bounds (the necessary terms will be defined in the talk). We will indicate the proofs of these statements, with improved bounds. The relevant constants are the roots of some cubics, square root of two, and the golden ratio.

2008-2009 seminar