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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.
      
We define a map from an affine Weyl group to the set of conjugacy classes of an ordinary Weyl group.
      
Open subsets of projective spaces with a good quotient by an action of a reductive 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.
      
Reductive group actions on affine quadrics with 1-dimensional quotient: Linearization when a linear model exists
      
We study reductive group actions on complex affine quadrics.
      
We prove that if a reductive group action on an affine quadric with a 1-dimensional quotient has a linear model, then the action is linearizable.
      
We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group.
      
Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations.
      
LetH? be a real form of a complex semisimple group.
      
Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants.
      
Methods are developed for the calssification of homogeneous Riemannian hypersurfaces and the classification of linear transitive reductive algebraic group actions on pseudo-Riemannian hypersurfaces.
      
In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues.
      
This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
      
We prove the following result: LetG be a finite irreducible linear group.
      
If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1).
      
In the interesting case when the group is of Coxeter typeDn (n≥4) we use higher polarization operators introduced by Wallach.
      
We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of typeF4.
      
TheS-arithmetic group г is of typeFn, resp.FPn, if and only if for allp inS thep-adic completionGp of the corresponding algebraic groupG is of typeCn resp.CPn.
      
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.
      
 

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