Abstract. The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and p-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of R n) are generalized to the S-arithmetic setting. 0.1. Actions of unipotent subgroups on homogeneous spaces of real Lie groups provide examples of important dynamical systems with numerous applications to geometry and number theory. A fundamental phenomemon discovered by G. A. Margulis in 1971 [M] showed that, in sharp contrast to partially hyperbolic flows, orbits of unipotent flows

The automorphism group of the Barnes-Wall lattice Lm in dimension 2m (m ̸ = 3) is a subgroup of index 2 in a certain “Clifford group ” of structure 2 1+2m +.O+(2m, 2). This group and its complex analogue CIm of structure (2 1+2m + YZ8).Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge’s 1996 result that the space of invariants for Cm of degree 2k is spanned by the complete weight enumerators of the codes C ⊗ F2m, where C ranges over all binary self-dual codes of length 2k; these are a basis if m ≥ k − 1. We also give new constructions for Lm and Cm: let M be the Z [ √ [ 2]-lattice with Gram matrix

0.1. We define the relation of mirror symmetry on the class of pairs (complex abelian variety A + an element of the complexified ample cone of A) and study its properties. More precisely, let A be a complex abelian variety, C a A ⊂ NSA(R) – the ample cone of A and put

Abstract. For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map RG: H 1 (F, G) → H 3 (F, Q/Z(2)). This includes the Arason invariant for quadratic forms and Rost’s mod 3 invariant for Albert algebras as special cases. We show that RG has trivial kernel if G is quasi-split of type E6 or E7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank. For G an almost simple simply connected algebraic group over a field F, the set of all natural transformations of functors H 1 (?, G) − → H 3 (?, Q/Z(2)) is a finite cyclic group [KMRT98, §31] with a canonical generator. (Here Hi (?, M) is the Galois cohomology functor which takes a field extension of your base field F and returns a group if M is abelian and a pointed set otherwise. When F has characteristic zero, Q/Z(2) is defined to be lim → µ ⊗2 n for µ n the algebraic groups of nth roots of unity; see [KMRT98, p. 431] or [Gilb, I.1(b)] for a more complete definition.) This generator is called the Rost

a are commensurable with each other and j n a j = [ : \a 1 a], the set n a is nite for a 2 G(Q ). The cardinality of this set will be denoted by deg(a). If a = q deg(a) i=1 a i for some a i 2 G(Q ), then T a (f)(x) = 1 deg(a) deg(a) X i=1 f(a i x): In particular, the above expression is independent of the choice of a i 's. Note also that the operator norm of T a is precisely 1 du

1.1. A fake projective plane is a smooth compact complex surface P which is not the complex projective plane but has the same first and second Betti numbers as the complex projective plane (b1(P) = 0 and b2(P) = 1). It is well-known that such a surface is projective algebraic and is the quotient of the complex two ball in C2 by a cocompact torsion-free discrete subgroup of PU(2, 1). These are surfaces with the smallest Euler-Poincaré characteristic among all smooth surfaces of general type. The first fake projective plane was constructed by Mumford [Mu] using p-adic uniformization, and more recently, two more examples were found by related methods by Ishida-Kato in [IK]. We have just learnt from Keum that he has an example which may be different from the earlier three. A natural problem in complex algebraic geometry is to determine all fake projective planes. It is proved in [Kl] and [Y] that the fundamental group of a fake projective plane, considered as a lattice of PU(2, 1), is arithmetic. In this paper we make use of this arithmeticity result and the volume formula of [P], together with some number theoretic estimates, to make a complete list of all fake projective planes,

Abstract. We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank ≤ 2 · dimH. As an application, we obtain a new and elementary proof of a conjecture of Connes and Sullivan on amenable actions, which was first proved by Zimmer. 1.

We show that the representation variety for the surface group in characteristic zero is (absolutely) irreducible and rational over Q. Visiting the University of Michigan (Ann Arbor, MI 48109, USA) in 1992-94. Introduction Let \Gamma be a finitely generated group. For any algebraic group G the set R(\Gamma; G) of all representations ( = homomorphisms) ae : \Gamma ! G is known to have a natural structure of an algebraic variety, and endowed with this structure is called the variety representations of \Gamma in G (cf. [Lu-M], [Pl-R]). In the case G = GL n which is analyzed by the classical representation theory, R(\Gamma; GL n ) is denoted simply by R n (\Gamma) and called the variety of n- dimensional representations of \Gamma. Since R n (\Gamma) is defined by the equations arising from the relations for the generators of \Gamma, a special role in this theory is played by the one-relator groups \Gamma =! x 1 ; : : : ; x n j r = 1 ? : The methods of this paper allow to consider in ...

We study subgroups G of GL(n, R) definable in o-minimal expansions M = (R, +, , ...) of a real closed field R. We prove several results such as: (a) G can be defined using just the field structure on R together with, if necessary, "power functions", or an "exponential function" definable in M . (b) If G has no infinite, normal, definable abelian subgroup, then G is semialgebraic. We also characterize the definably simple groups definable in o-minimal structures as those groups elementarily equivalent to simple Lie groups, and we give a proof of the Kneser-Tits conjecture for real closed fields. # Supported by NSF grant. + Supported by NSF grant 1 1 Introduction In most of this paper, R will be a real closed field and M will be an o- minimal expansion of (R, <, +, ). "Definable" means definable in M . We are interested in definable groups G < GL(n, R). (This is what we mean by a linear group definable in an o-minimal structure.) We are are interested in questions of th...

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