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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 theirq-analogs in the representationsVnλ,n∈?.
      
As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnS-arithmetic 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 above-mentioned 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 semi-invariants span the space of all semi-invariants for any quiver and any infinite base field.
      
We give the classification of all finite dimensional Levi-Tanaka algebras of CR codimension two and construct the corresponding standard homogeneous CR manifolds.
      
We describe all pairs (G, H) such that, for any affineG-varietyX with a denseG-orbit isomorphic toG/H, the number ofG-orbits 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.
      
 

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