On properly essential classical conformal diffeomorphism groups

In this article, we prove that various classical conformal diffeomorphism groups, which are known to be essential (Banyaga, J Geom 68(1–2):10–15, 2000), are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological equation. Furthermore, we study the orbit of a tensor field under the action of the conformal diffeomorphism group for these classical conformal structures. On every closed contact manifold, we find conformal contact forms that are not diffeomorphic.     

The rank of planar discontinuous groups

In [3] H. Zieschang develops the Nielsen method for amalgamated products and uses this to compute the rank of planar discontinuous groups, a originally overlooked case was treated in [1]. In this note we give a short proof of this result using the Nielsen method for groups acting on trees [2].     

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, ${{\rm Sp}(2,\mathbb {R})}Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of K?hler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.     

Regular unipotent elements from subsystem subgroups of rank 2 in modular representations of classical groups

Images of regular unipotent elements from subsystem subgroups of type A₂ and B₂ in irreducible modular representations of classical groups are studied. For images of such elements and representations with locally small highest weights, all sizes of Jordan blocks of one and the same parity are found. Bibliography: 17 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 159–178.     

The representation-theoretic rank of the doubles of quasi-quantum groups

We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra H and of its quantum double D(H), within the rigid braided category of finite dimensional left D(H)-modules.     

Generation of classical groups

Each finite simple group other thanU ₃(3) can be generated by three of its involutions. In fact, each such group is generated by two elements, of which one is strongly real and the other is an involution.The authors thank the MSRI in Berkeley, where this work was begun in Fall 1990, for its hospitality. The first and third author gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft.     

The Chern character for classical matrix groups

We give explicit formulas for representations of classical matrix groups whose Chern characters have lowest order terms equal to standard characteristic classes. For , the Euler class does not arise in this way, but does arise in this way.

     

Weyl groups of small groups of finite Morley rank

We examine Weyl groups of minimal connected simple groups of finite Morley rank of degenerate type. We show that they are cyclic, and lift isomorphically to subgroups of the ambient group.     

Solvable groups of finite non-Abelian rank

The concept of the non-Abelian rank of a group is introduced. Solvable groups of finite non-Abelian rank are studied and it is proved that their (special) rank is finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 159–164, February, 1990.