A generalization of Serre’s condition $$\mathrm {(F)}$$ ( F ) with applications to the finiteness of unramified cohomology

Mathematische Zeitschrift - In this paper, we introduce a condition ( $$\mathrm {F}_m'$$ ) on a field K, for a positive integer m, that generalizes Serre’s condition (F) and which still...     

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let be a finite dimensional division algebra having center , and let be a normal subgroup of finite index. Suppose is not solvable. Then we may assume that is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property . This property includes the requirement that the diameter of the commuting graph should be , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of has Property , then is open with respect to a nontrivial height one valuation of (assuming without loss of generality, as we may, that is finitely generated). After establishing the openness of (when is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of over its prime subfield to eliminate as a possible quotient of , thereby obtaining a contradiction and proving our main result.

     

Structure of groups of rational points of classical algebraic groups over number fields

Dedicated to the memory of A. I. Mal'tsev.     

Representation varieties of the fundamental groups of compact orientable surfaces

We show that the representation variety for the surface group in characteristic zero is (absolutely) irreducible and rational over ℚ. This work was supported in part by the International Science Foundation (Grant No. MWQ000). Visiting the University of Michigan (Ann Arbor, MI 48109, USA) in 1992–94.     

On the finite dimensional unitary representations of Kazhdan groups

We use A. Weil's criterion to prove that all finite dimensional unitary representations of a discrete Kazhdan group are locally rigid. It follows that any such representation is unitarily equivalent to a unitary representation over some algebraic number field.

     

Algebraic groups

This paper is a continuation of V. P. Platonov's survey in Vol. 11 ofAlgebra, Topology, Geometry. It consists of two parts, the first of which deals with structural problems and rationality questions for algebraic groups, and the second with the arithmetic theory of algebraic groups. Particular attention is paid to properties of semisimple groups and their groups of rational points.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 21, pp. 80–134, 1983.     

Valuation-like maps and the congruence subgroup property

Let D be a finite dimensional division algebra and N a subgroup of finite index in D ^×. A valuation-like map on N is a homomorphism ϕ:N?Γ from N to a (not necessarily abelian) linearly ordered group Γ satisfying N _<-α+1⊆N _<-α for some nonnegative α∈Γ such that N _<-α≠=?, where N _<-α={x∈N|ϕ(x)<-α}. We show that this implies the existence of a nontrivial valuation v of D with respect to which N is (v-adically) open. We then show that if N is normal in D ^× and the diameter of the commuting graph of D ^×/N is ≥4, then N admits a valuation-like map. This has various implication; in particular it restricts the structure of finite quotients of D ^×. The notion of a valuation-like map is inspired by [27], and in fact is closely related to part (U3) of the U-Hypothesis in [27]. Oblatum 14-VII-2000 & 22-XI-2000?Published online: 5 March 2001     

On abstract representations of the groups of rational points of algebraic groups in positive characteristic

We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg’s perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups.     

Low-dimensional linear representations of

We classify all complex representations of the automorphism group of the free group of dimension Among those representations is a new representation of dimension which does not vanish on the group of inner automorphisms.