The *Amitsur Memorial Symposium* is an annual conference in memory of Prof. Shimshon Avraham Amitsur. It is hosted by a different institution each year.

The 25th *Amitsur Memorial Symposium *will be held at the Einstein Institute of Mathematics, the Hebrew University of Jerusalem.

## Speakers, titles and abstracts

**Yael Algom-Kfir**

Title:** ****The completion of a non-symmetric metric space**

Abstract: The Teichmuller space with the Thurston metric and Outer Space with the Lipschitz metric are two examples of spaces with an asymmetric metric i.e. d(x,y) \neq d(y,x). The latter case is also incomplete: There exist Cauchy sequences that do not have a limit. We develop the theory of the completion of an asymmetric space and give lots of examples. Time permitting we will describe the case of Outer Space.

**Amiram Braun**

Title: **The polynomial question in modular invariant theory, old and new**.

Abstract: Let G be a finite group, V a finite dimensional G- module over a field F, and S(V) the symmetric algebra of V. The above problem seeks to determine when is the ring of invariants S(V)^G , a polynomial ring. In the non-modular case (i.e. char(F) being prime to order(G)), this was settled in the Shephard-Todd-Chevalley theorem. The modular case (i.e. char(F) divides order (G) ), is still wide open. I shall discuss some older results due to Serre, Nakajima , Kemper-Malle and explain some new results, mostly in dimension 3.

**Tsachik Gelander**

Title: **Local rigidity of uniform lattices**

Abstract. We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in Isom(X) where X is a proper CAT(0) space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang’s finiteness theorem for certain non-positively curved metric spaces.

This is a joint work with Arie Levit.

**Lev Glebsky**

Title: **Approximations of groups by finite and linear groups**.

Abstract: The sofic groups and hyperlinear groups are groups approximable by finite symmetric

and by unitary groups, respectively. I recall their definitions and discuss why those classes of groups

are interesting. Then I consider approximations by other classes of groups and review some results,

including rather recent ones by N. Nikolov, J. Schneider, A.Thom, https://arxiv.org/abs/1703.06092 .

If time permits I'll speak about stability and its relations with approximability.

**Arye Juhasz**

Title: **On the center of Artin groups**.

Abstract: Let A be an Artin group. It is known that if A is spherical (of finite type) and irreducible (not a direct sum), then it has infinite cyclic center.

It is conjectured that all other irreducible Artin groups have trivial center. I prove this conjecture under a stronger assumption that not being spherical namely, if there is a standard generator which is not contained in any 3-generated spherical standard parabolic subgroup. The main tool is relative presentations of Artin groups.

**Alex Lubotzky**

Title: ** First order rigidity of high-rank arithmetic groups**

Abstract: The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics.

It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.

A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.

We will talk about a new type of rigidity : "first order rigidity". Namely if G is such a non-uniform characteristic zero arithmetic group and H a finitely generated group which is elementary equivalent to it then H is isomorphic to G.

This stands in contrast with Zlil Sela's seminal work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them.

Joint work with Nir Avni and Chen Meiri.

**Chloe Perin**

Title: **Forking independence in the free group**

Abstract: Model theorists define, in structures whose first-order theory is "stable" (i.e. suitably nice), a notion of independence between elements. This notion coincides for example with linear independence when the structure considered is a vector space, and with algebraic independence when it is an algebraically closed field. Sela showed that the theory of the free group is stable. In a joint work with Rizos Sklinos, we give an interpretation of this model theoretic notion of independence in the free group using Grushko and JSJ decompositions.

**Elyiahu Rips**

Title: **Free Engel groups**

Abstract: A free n-Engel group is the relatively free group of the variety of groups with the identical relation [x, y, y,...,y (n times)]=1. Let n>=20. We show that the free Engel group on at least two generators is not locally nilpotent. Our approach to Engel groups combines

geometric and combinatorial methods. On the geometric side, we consider graded van Kampen diagrams, and we prove that they display (discrete) negative curvature properties. To do this, we construct a canonical form of elements in each consecutive rank (this is the combinatorial aspect). Using the canonical form, we obtain "parallel meetings" between the regions of higher ranks of the graded van Kampen diagram and, using surgery, improve it to direct meeting. The combinatorial structure of the relators secures that this direct meeting is (relatively) short. Given the structure of graded van Kampen diagrams, we deduce a graded version of Greendlinger's Lemma and then establish the properties of the group.

Joint work with Arye Juhasz

**Malka Schaps**

Title: **Symmetric Kashivara crystals of type A in low rank**

Abstract: The basis of elements of the highest weight representations of affine Lie algebra of type A can be labeled in three different ways, my multipartitions, by piecewise linear paths in the weight space, and by canonical basis elements. The entire infinite basis is recursively generated from the highest weight vector of operators f_i from the Chevalley basis of the affine Lie algebra, and organized into a crystal called a Kashiwara crystal. We describe cases where one can move between the different labelings in a non-recursive fashion, particularly when the crystal has some symmetry.

Joint work with Ola Amara-Omari

**Aner Shalev**

Title: **The length and depth of finite groups, algebraic groups and Lie groups**

Abstract: The length of a finite group G is defined to be the maximal length of an unrefinable chain of subgroups going from G to 1. This notion was studied by many authors since the 1940s.

Recently there is growing interest also in the depth of G, which is the minimal length of such a chain. Moreover, similar notions were defined and studied for important families of infinite groups, such as connected algebraic groups and connected Lie groups.

I will describe recent works on these topics, joint with Tim Burness and Martin Liebeck. The proofs use

a variety of tools, including recent results in analytic number theory.

## Programme

**Tuesday, June 26th**:

9:30 -- Gathering

10:00 -- Lubotzky

11:00 -- Coffee break

11:30 -- Schaps

12:30 -- Lunch break

14:00 -- Juhasz

15:00 -- Shalev

16:00 -- Coffee break

16:30 -- Glebsky

The **symposium dinner** will be held at 7pm at the **Piccolino** restaurant:

Yoel Moshe Solomon Street No.12, Jerusalem

Tel. 02-6244186

The dinner will be attended by Amitsur's children.

**Wednesday, June 27th**:

10:00 -- Rips

11:00 -- Coffee break

11:30 -- Braun

12:30 -- Lunch break

14:00 -- Perin

15:00 -- Gelander

16:00 -- Coffee break

16:30 -- Algom-Kfir.