We construct a finitely presented torsion-free simple group Σ0, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2, 4]). We refine their methods and get Σ0 as an index 4 subgroup of a group Σ < Aut(T12)×Aut(T8) presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4,...

We show that any non-zero orbit under a non-compact, simple, irreducible linear group is dense in the Bohr compactification of the ambient space.

Let G be a connected Lie group acting by algebra automorphisms on a finite-dimensional complex central simple algebra A. The algebra A is isomorphic to the endomorphism algebra of a projective representation V of G. We study the invariant subalgebras of A. In particular, we show that if V is irreducible, then the invariant subalgebras appear in dual pairs arising from factorizations of V . We a...

Simple finite group schemes over an algebraically closed field of positive characteristic p 6= 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of height one corresponding to the restricted simple Lie algebras.

We present a new non perturbative renormalization group for classical simple fluids. The theory is built in the Grand Canonical ensemble and in the framework of two equivalent scalar field theories as well. The exact mapping between the three renormalization flows is established rigorously. In the Grand Canonical ensemble the theory may be seen as an extension of the Hierarchical Reference Theo...

Thisnotedescrib esrecen tresearc h usingstructural propertiesofflnitegroupstodevise–cien talgorithms forgroupcomputation. 1991Mathematics SubjectClassiflcation: 20B40,20D05,68Q40 KeywordsandPhrases: grouptheoretic algorithms, implegroups

We show that Thompson’s group F is the symmetry group of the ‘generic idempotent’. That is, take the monoidal category freely generated by an object A and an isomorphism A ⊗ A A; then F is the group of automorphisms of A.

Let G be a group, F a field, and A a finite-dimensional central simple algebra over F on which G acts by F -algebra automorphisms. We study the ideals and subalgebras of A which are preserved by the group action. Let V be the unique simple module of A. We show that V is a projective representation of G and A = EndD(V ) makes V into a projective representation. We then prove that there is a natu...