G. A. Margulis

Flows on homogeneous spaces and Diophantine approximation on manifolds

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.

Logarithm laws for flows on homogeneous spaces

Abstract.In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {At | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements ft of G under which #{t∈ℕ | ftx∈At} is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov.

Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions

Notation. The main objects of this paper are n-tuples y = (y1, . . . , yn) of real numbers viewed as linear forms, i.e. as row vectors. In what follows, y will always mean a row vector, and we will be interested in values of a linear form given by y at integer points q = (q1, . . . , qn)T , the latter being a column vector. Thus yq will stand for y1q1 + · · ·+ ynqn. Hopefully it will cause no confusion. We will study differentiable maps f = (f1, . . . , fn) from open subsets U of R to R; again, f will be interpreted as a row vector, so that f(x)q stands for q1f1(x) + · · · + qnfn(x). In contrast, the elements of the “parameter set” U will be denoted by x = (x1, . . . , xd) without boldfacing, since the linear structure of the parameter space is not significant. For f as above we will denote by ∂if : U 7→ R, i = 1, . . . , d, its partial derivative (also a row vector) with respect to xi. If F is a scalar function on U , we will denote by∇F the column vector consisting of partial derivatives of F . With some abuse of notation, the same way we will treat vector functions f : namely, ∇f will stand for the matrix function U 7→ Md×n(R) with rows given by partial derivatives ∂if . We will also need higher order differentiation: for a multiindex β = (i1, . . . , id), ij ∈ Z+, we let |β| = ∑d j=1 ij and ∂β = ∂ i1 1 ◦ · · · ◦ ∂ id d . Unless otherwise indicated, the norm ‖x‖ of a vector x ∈ R (either row or column vector) will stand for ‖x‖ = max1≤i≤k |xi|. In some cases however we will work with the Euclidean norm ‖x‖ = ‖x‖e = √∑k i=1 x 2 i , keeping the same notation. This distinction will be clearly emphasized to avoid confusion. We will denote by R1 the set of unit vectors in R (with respect to the Euclidean norm). We will use the notation |〈x〉| for the distance between x ∈ R and the closest integer, |〈x〉| def = mink∈Z |x− k|. (It is quite customary to use ‖x‖ instead, but we are not going to do this in order to save the latter notation for norms in vector spaces.) If B ⊂ R, we let |B| stand for the Lebesgue measure of B.

Discrete Subgroups and Ergodic Theory

Publisher Summary This chapter focuses on the discrete subgroups and the ergodic theory. It presents the formulation of results and conjectures. The results on the behavior of “individual” orbits and the description of invariant measures for actions of groups on homogeneous spaces have been provided. Some number theoretic corollaries of these results have also been provided. The chapter proves Oppenheims conjecture on values of indefinite quadratic forms.

The Differential of a Quasi-Conformal Mapping of a Carnot-Caratheodory Space

The theory of quasi-conformal mappings has been used to prove rigidity theorems on hyperbolic n space over the division algebras ℝ, ℂ, ℍ, and \({\Bbb O}\), by studying quasi-conformal mappings on their boundary spheres S kn−1 at infinity, where k is the dimension of the division algebra. The notion of quasiconformal mappings for such spaces, first introduced in [Mo2]w, as subsequently reformulated by Pansu in terms of Carnot-Caratheodory spaces M, and Pansu studied quasi-conformal mappings for the special case of graded nilpotent groups M. Subsequently, Pansu’s definition was simplified in [Mo3], and this simpler definition was employed by Koranyi-Reimann in their study of quasi-conformal mappings of the nilpotent Heisenberg group operating on the boundary of complex hyperbolic n-space and transitive on the complement of one point.

Semigroups containing proximal linear maps

A linear automorphism of a finite dimensional real vector spaceV is calledproximal if it has a unique eigenvalue—counting multiplicities—of maximal modulus. Goldsheid and Margulis have shown that if a subgroupG of GL(V) contains a proximal element then so does every Zariski dense subsemigroupH ofG, providedV considered as aG-module is strongly irreducible. We here show thatH contains a finite subsetM such that for everyg∈GL(V) at least one of the elements γg, γ∈M, is proximal. We also give extensions and refinements of this result in the following directions: a quantitative version of proximality, reducible representations, several eigenvalues of maximal modulus.

Free subgroups of the homeomorphism group of the circle

Abstract Tits alternative says that a finitely generated linear group either contains a non-commutative free subgroup or is virtually solvable. It is known ( see [2]) that an analogue of Tits alternative is not true for subgroups of the group Homeo(S 1 ) of all homeomorphisms of the circle S 1 and even for subgroups of the group of C ∞ -diffeomorphisms of S 1 . The main purpose of this Note is to prove a conjecture of E. Ghys which can be viewed as a replacement of Tits alternative for Homeo(S 1 ) and which says that if G is a subgroup of Homeo(S 1 ) containing no free non-commutative subgroup then there is a G -invariant probability measure on S 1 .

Aperiodic tilings of the hyperbolic plane by convex polygons

Several aperiodic hyperbolic tiling systems consisting of a single convex tile are constructed.

Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices

Under some weak hyperbolicity conditions, we establish C^0 - and C^\infty -local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0 -local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0 -local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmers cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty -local rigidity is deduced from the C^0 -local rigidity following a procedure outlined by Katok and Spatzier. Using similar considerations, we also establish C^0 -global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

Singular systems of linear forms and non-escape of mass in the space of lattices

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.