The following results are proved: (1) The center of any finite index subgroup of an irreducible, infinite, non-affine Coxeter group is trivial; (2) Any finite index subgroup of an irreducible, infinite, non-affine Coxeter group cannot be expressed as a product of two nontrivial subgroups. These two theorems imply a unique decomposition theorem for a class of Coxeter groups. . MSC 2000 Subject C...

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn−1. This provides a new example of an infinite type Artin group which inj...

In this note, we study the existence and structure of the homogeneous space G/H for algebraic groups H ⊂ G. Let k be a field. All schemes considered will be k-schemes. By an affine algebraic group, we mean an affine group scheme of finite type over k. Note that we do not assume our schemes are reduced yet. We will only consider affine algebraic groups. From now on, G will denote an algebraic gr...

We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study the properties of the action of the homotopy mapping class groups on deformatio...

We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v]⋊Γ). We ...

We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v] o Γ). W...

We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.

We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic , then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be torsion fre...