On discrete subgroups containing a lattice in a horospherical subgroup

We showed in [Oh] that for a simple real Lie groupG with real rank at least 2, if a discrete subgroup Γ ofG contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice inG, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horospherical subgroup, provided Γ is Zariski dense inG.     

Superrigidity of lattices in solvable Lie groups

Summary Let be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that Ad _G has the same Zariski closure as AdG. If : GL _n () is any finite-dimensional representation of , we show that virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if is isomorphic to a closed, cocompact subgroup of another simply connected, solvable Lie groupG, then any isomorphism from to extends to a crossed isomorphism fromG toG. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.Oblatum 5-VII-1994 & 15-IV-1995     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

Compactness of minimal closed invariant sets of actions of unipotent groups

Let G be a connected Lie group, let Γ be a lattice in G, and let be a unipotent subgroup of G. It is proved that, for the natural action of on G/Γ, every minimal closed -invariant subset is compact. Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday     

Distribution of lattice orbits on homogeneous varieties

Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H\ G. For an increasing family of finite subsets {Γ _T : T > 0}, a dense orbit υ· Γ, υ∈V and compactly supported function φ on V, we consider the sums . Understanding the asymptotic behavior of S _φ,υ (T) is a delicate problem which has only been considered for certain very special choices of H,G and {Γ _T }. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have where G _T = {g ∈G: ||g|| < T} and Γ _T = G _T ∩ Γ. We also show that the asymptotics of S _{φ, υ} (T) is governed by where ν is an explicit limiting density depending on the choice of υ and || · ||. Submitted: March 2005 Revision: April 2006 Accepted: June 2006     

Certain inverse problems for parabolic equations

In the paper, we study the inverse problem of finding the solution u and the coefficient q from the following data:

Ergodicity of generalised Patterson-Sullivan measures in higher rank symmetric spaces

Let X = G/K be a higher rank symmetric space of noncompact type and a discrete Zariski dense group. In a previous article, we constructed for each G-invariant subset of the regular limit set of Γ a family of measures, the so-called (b, Γ · ξ)-densities. Our main result here states that these densities are Γ-ergodic with respect to an important subset of the limit set which we choose to call the ``ray limit set'. In the particular case of uniform lattices and products of convex cocompact groups acting on the product of rank one symmetric spaces every limit point belongs to the ray limit set, hence our result is most powerful for these examples. For nonuniform lattices, however, it is a priori not clear whether the ray limit set has positive measure with respect to a (b, Γ · ξ)-density. Using a counting theorem of Eskin and McMullen, we are able to prove that the ray limit set has full measure in each G-invariant subset of the limit set.     

Spectra of ergodic group actions

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ^∞-spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim sup_g→∞Re (g)<1, where is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim sup_g→∞Re (g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .     

Up–down representations and ergodic theory in nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G.     

Limit distributions of expanding translates of certain orbits on homogeneous spaces

LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and . LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U⁺ ={g&#x03B5;G:a ⁻ⁿ gaⁿ} →e as n → ∞. Let Ω be a non-empty open subset ofU ⁺ andn _i → ∞ be any sequence. It is showed that . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.     

Pseudo algebraically closed fields over rings

We prove that for almost allσ ∈G ℚ the field has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→ there exists a point a ∈ such thatϕ(a) ∈ ℤ^r. We then say that is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.     

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

On the maximum modulus principle for polynomials in a quasidisk

LetG⊂C be a quasidisk,K ⊂ G be a compact set, andp _n be a non-constant complex polynomial of degree at mostn. We establish the inequality whereα _n < 0 depends onn, K, and the geometrical structure of ϖG.     

Estimates for the number of rational points on convex curves and surfaces

Let Γ ⊂ ℝ^d be a bounded strictly convex surface. We prove that the number k_n(Γ) of points of Γ that lie on the lattice satisfies the following estimates: lim inf k_n(Γ)/n^d−2 < ∞ for d ≥ 3 and lim inf k_n(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189.     

Kernels of Toeplitz operators,smooth functions,and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle and let K_p(ϕ) denote the kernel of the Toeplitz operator T_ϕ in the Hardy space H^p, p≥1; . Suppose K_p(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in K_p(ϕ). One of the main results is as follows. Theorem 1 Let 1<p, q<+∞, 1<r≤+∞, q⁻¹=p⁻¹+r⁻¹. Suppose |ϕ|≡1 on and ϕ∈W _r ¹ (i.e., ). Then K_p(ϕ)⊂W _q ¹ . Moreover, for any f∈K_p(ϕ) we have ‖f′‖_q≤c(p, r)‖ϕ′‖_r ‖f‖. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21. Translated by K. M. D'yakonov.     

Restriction De La Cohomologie D’une Variété De Shimura à Ses Sous-variétés

Let G be a connected semisimple group over . Given a maximal compact subgroup K ⊂ G() such that X = G()/K is a Hermitian symmetric domain, and a convenient arithmetic subgroup Γ ⊂ G(), one constructs a (connected) Shimura variety S = S(Γ) = Γ\X. If H ⊂ G is a connected semisimple subgroup such that H() / K is maximal compact, then Y = H()/K is a Hermitian symmetric subdomain of X. For each g ∈ G() one can construct a connected Shimura variety S(H, g) = (H() ∩ g ⁻¹Γg)\Y and a natural holomorphic map j _g : S(H, g) → S induced by the map H() → G(), h → gh. Let us assume that G is anisotropic, which implies that S and S(H, g) are compact. Then, for each positive integer k, the map j _g induces a restriction map

On the existence of spines for ${\mathbb{Q}}${\mathbb{Q}}-rank 1 groups

Let X = Γ \G/ K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has -rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D₀. We prove that D₀ is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ -conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ.