Wiener Tauberian theorem for rank one semisimple Lie groups and for hypergeometric transforms

We prove a genuine analogue of the Wiener Tauberian theorem for , where G is a real rank one noncompact, connected, semisimple Lie group with finite centre. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm, and Y. Weit. We extend this result for hypergeometric transforms and as an application we prove an analogue of Furstenberg theorem on harmonic functions for hypergeometric transforms.     

A (c. L*-Geometry for the Sporadic Group J2

We prove the existence of a rank three geometry admitting the Hall–Janko group J2 as flag-transitive automorphism group and Aut(J2) as full automorphism group. This geometry belongs to the diagram (c·L*) and its nontrivial residues are complete graphs of size 10 and dual Hermitian unitals of order 3.     

An infinite family of simply connected flag-transitive tilde geometries

We consider tilde-geometries (orT-geometries), which are geometries belonging to diagrams of the following shape: Here the rightmost edge stands for the famous triple cover of the classical generalized quadrangle related to the group Sp₄(2). The automorphism group of the cover is the nonsplit extension 3·Sp₄(2) – 3 ·S ₆. Five examples of flag-transitiveT-geometries were known. These are rank 3 geometries related to the groupsM ₂₄ (the Mathieu group),He (the Held group) and and 3⁷·Sp₆(2) (a nonsplit extension); a rank 4 geometry related to the Conway groupCo ₁ and a rank 5 geometry related to the Fischer-Griess Monster groupF ₁. In the present paper we construct an infinite family of flag-transitiveT-geometries and prove that all the new geometries are simply connected. The automorphism group of the rankn geometry in the family is a nonsplit extension of a 3-group by the symplectic group Sp_2n (2). The rank of the 3-group is equal to the number of 2-dimensional subspaces in ann-dimensional vector space over GF(2).     

A design and a geometry for the group Fi 22

The Fischer group Fi ₂₂ acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it.      

On the Generation of Some Embeddable GF(2) Geometries

The generating rank is determined for several GF(2)-embeddable geometries and it is demonstrated that their generating and embedding ranks are equal. Specifically, we prove that each of the two generalized hexagons of order (2, 2) has generating rank 14, that the central involution geometry of the Hall-Janko sporadic group has generating rank 28, and that the dual polar space DU(6,2) has generating rank 22. We also include a survey of all instances in which either the generating or embedding rank of an embeddable GF(2) geometry is known.     

AF Embeddability of Crossed Products of AT Algebras by the Integers and Its Application

We will show that the crossed products of unital simple real rank zero AT algebras by the integers are AF embeddable. This is a generalization of Brown's AF embedding theorem. As an application, we will prove the AF embeddability of crossed product algebras arising from certain minimal dynamical systems induced by two commuting homeomorphisms.     

Fixed points of involutive automorphisms of the Bruhat order

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein^* over Z₂ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right.     

A new example of a uniformly Levi degenerate hypersurface in C<Superscript>3</Superscript>

We present a homogeneous real analytic hypersurface in C³, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube Γ_C over the two-dimensional real cone in R³ is also homogeneous and has a seven-dimensional CR automorphism group. However, our example isnot biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of Γ_C are, in contrast, noncommutative.     

Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups

We prove that there exists an algorithm which solves a conjugacy problem for finite subgroups in automorphism and outer automorphism groups of a free group of finite rank. Of independent interest is the construction of an algorithm of decomposing an arbitrary free-by-finite group into a fundamental group of a finite graph of finite groups, with the number of steps evaluated explicitly. In passing, we solve the conjugacy problem for finite subgroups in almost free groups. As a consequence, an algorithm is obtained computing generating sets for a group of fixed points in an arbitrary finite automorphism group of a free group of finite rank.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 558–606, September-October, 1995.Supported by the RFFR grant No. 93-011-1508 and by the ISF (International Science Foundation) grant RPC000.     

Non-discrete affine buildings and convexity

Affine buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to ask for analogs of results for symmetric spaces. We prove a version of Kostant?s convexity theorem for thick non-discrete affine buildings. Kostant proves that the image of a certain orbit of a point x in the symmetric space under a projection onto a maximal flat is the convex hull of the Weyl group orbit of x. We obtain the same result for a projection onto an apartment in an affine building. The methods are mostly borrowed from metric geometry. Our proof makes no appeal to the automorphism group of the building. However the final result has an interesting application for groups acting nicely on non-discrete buildings, such as groups admitting a root datum with non-discrete valuation. Along the proofs we obtain that segments are contained in apartments and that certain retractions are distance diminishing.     

On a theorem by Farb and Masur

Farb and Masur showed that an irreducible lattice in a semisimple Lie group of rank at least two always has finite image by a homomorphism into the outer automorphism group of a closed, orientable surface group. We point out that their theorem extends to the outer automorphism groups of a certain class of torsion-free, freely indecomposable word-hyperbolic groups.

     

The automorphism group of a product of hypergraphs

A theorem of G. Sabidussi (1959, Duke Math. J. 26, 693–696) gives necessary and sufficient conditions for the automorphism group of the wreath product of two graphs to be the wreath product of their respective automorphism groups. In this paper we define a wreath product of hypergraphs and prove a theorem extending that of Sabidussi.     

Fixed points of automorphisms of Lie rings and locally finite groups

Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group C _P(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 706-723, November-December, 1995.Supported by RFFR grant No. 94-01-00048-a and by ISF grant NQ7000.     

CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ ⁿ⁺¹ admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ ⁿ⁺¹ with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group. Research of the first named author is partially supported by The Grant R01-2005-000-10771-0 of The Korea Science and Engineering Foundation.     

The Module Structure Theorem for Sobolev Spaces on Open Manifolds

We prove a general embedding theorem for Sobolev spaces on open manifolds of bounded geometry and infer from this the module structure theorem. Thereafter we apply this to weighted Sobolev spaces.     

Sporadic groups and automorphisms of linear spaces

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let 𝒮 be a finite linear space on which G acts as a line‐transitive automorphism group. Then the socle of G is not a sporadic group. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 353–362, 2000     

An Embedding Problem with Cyclic Kernel

The problem of embedding a quadratic extension of a number field into an extension with a cyclic 2-group is studied. We prove a reduction theorem showing that, under the compatibility condition, an additional embedding condition consists of the solvability of a problem with cyclic kernel of order 16 (of course, if the degree of the required field is no less than 16). An additional condition of embedding into a field of degree 16 is found; namely, the number generating the given quadratic extension must be a norm in a cyclotomic field containing the primitive eighth roots of unity. For Q, the embedding condition is simpler: all the odd prime divisors of the generating element must be congruent with 1 modulo the order of the extension group. In addition, the quadratic extension must be real. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 144–152.     

Indefinite Kähler submanifolds with positive index of relative nullity

We deal with complex submanifolds in indefinite space forms. In particular, submanifolds with large index of relative nullity are emphasized. In that context, we prove cylinder theorems in terms of indefinite metrics. We also give a systematic way of constructing a family of new complete and closed indefinite complex submanifolds in the projective setting.In the appendix, we show that the method used for complex cases can be applied to real indefinite geometry. We prove real cylinder theorems including B-scrolls in the general signature. We also show two decomposition lemmas which clarify the relationships between the Hartman-Nirenberg cylinder theorem and slanted cylinder theorems in indefinite geometry.     

FREE SOLVABLE P-ALGEBRAS

The construction of a free solvable P-algebra of finite degree k in the variety of all solvable P-algebras of degree at most k (k ≥ 1) has been given. Some properties of the same have been studied. The structure of the free solvable P-algebra has been viewed as a module over a ring with several objects. The Magnus embedding theorem associated with the Fox-derivative in a free group ring has been considered to prove properties associated with the partial (Fox) derivative in a free associative ring. Residual nilpotency and triviality of the center of a free metabelian P-algebra has been proved. Various properties of a homomorphism associated with a free metabelian P-algebra of finite rank have been studied. The non-embedding property of a free solvable P-algebra of degree k of higher rank in a lower rank has also been presented here.     

Rigidity of quantum tori and the Andruskiewitsch–Dumas conjecture

We prove the Andruskiewitsch–Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra ${\mathcal {U}}_q({\mathfrak {g}})$ of an arbitrary finite dimensional simple Lie algebra ${\mathfrak {g}}$ is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of ${\mathfrak {g}}$ and a torus of rank equal to the rank of ${\mathfrak {g}}$ . The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.