Andrei S. Rapinchuk

Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric spaces.

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.

Computation of the metaplectic kernel

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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 D be a finite dimensional division algebra having center K and let N ⊆ D× be a normal subgroup of finite index. Suppose D×/N is not solvable. Then we may assume that H := D×/N 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 (3 12 ). This property includes the requirement that the diameter of the commuting graph should be ≥ 3, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of D×/N has the property (312 ), then N is open with respect to a nontrivial height one valuation of D (assuming without loss, as we may, that K is finitely generated). After establishing the openness of N (when D×/N is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of K over its prime subfield, to eliminate H as a possible quotient of D×, thereby obtaining a contradiction and proving our main result.

Irreducible Tori in semisimple groups

Let G be an absolutely simple simply connected algebraic group over a global field k. In this note, we analyze arithmetic properties of the maximal k-tori of G .We establish density (in the variety of maximal tori) of the set of maximal k-tori which do not contain proper k-subtori (we call such tori k-irreducible). Using the fact that the k-irreducible tori we construct have the weak approximation property, we extend some of our previous results (contained in Publ. Math. Inst. Haute Etudes Sci. 84 (1996), 91-187, § 9) to global function fields. This allows one to establish the congruence subgroup property for the groups of points of G over semi-local subrings of k. Finally, we examine the strong approximation property in the maximal k-tori of G with respect to generalized arithmetic progressions.Irreducible Tori in semisimple groups (Erratum) dx.doi.org/10.1155/S1073792802112074

Division algebras with the same maximal subfields

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.

Bounded generation of some S-arithmetic orthogonal groups

Abstract Let f be a nondegenerate quadratic form in n⩾5 variables and of Witt index ⩾2 over a number field K , S be a finite set of places of K containing all archimedean places. We prove that the S -arithmetic group SO n (f) O (S) has bounded generation.

Normal subgroups ofSL 1,D and the classification of finite simple groups

LetD be a division algebra of degree three over an algebraic number fieldK and let G = SLD. We prove that the normal subgroup structure of G(K) is given by the Platonov-Margulis conjecture. The proof uses the classification of finite simple groups.

The genus of a division algebra and the unramified Brauer group

Let