all 
In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.


The aim of the paper is to describe all open subsets of a projective space with an action of a reductive group which admits a good quotient.


This allows us to obtain a complete list of all irreducible linear groups with a polynomial ring of invariants.


Whenever the action of a maximal torus on the coneCλ* has some nice properties, we obtain simple closed formulas for all weight multiplicities and theirqanalogs in the representationsVnλ,n∈?.


As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnSarithmetic subgroup of a semisimple group has all the finiteness propertiesFn.


We construct essentially all the irreducible modules for the multiparameter quantum function algebraF?φ[G], whereG is a simple simply connected complex algebraic group, and ? is a root of unity.


Lichtenstein in the caseu =(n, ?) or(n?), we prove that ?(q).ζ1/2 is non zero for all harmonic polynomialsq ∈S() \ {0}.


We classify all instances when a parabolic subgroupP ofG acts on its unipotent radicalPu, or onpu, the Lie algebra ofPu, with only a finite number of orbits.


We determine the covolumes of all hyperbolic Coxeter simplex reflection groups.


This family includes all modular representations of cyclic groups.


For a smooth oriented surface Σ, denote byM(Σ) the set of all ways to represent Σ as a result of gluing together standard spheres with holes ("the Lego game").


The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type.


For example, if our groupG isSn, these objects are field extensions; ifG=On, they are quadratic forms; ifG=PGLn, they are division algebras (all of degreen); ifG=G2, they are octonion algebras; ifG=F4, they are exceptional Jordan algebras.


We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the abovementioned examples.


In this paper we prove the dimension and the irreduciblity of the variety parametrizing all pairs of commuting nilpotent matrices.


We prove that these determinantal semiinvariants span the space of all semiinvariants for any quiver and any infinite base field.


We give the classification of all finite dimensional LeviTanaka algebras of CR codimension two and construct the corresponding standard homogeneous CR manifolds.


We describe all pairs (G, H) such that, for any affineGvarietyX with a denseGorbit isomorphic toG/H, the number ofGorbits inX is finite.


We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in ?(G) are diagonalizable.


We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection.

