On the Patterson–Sullivan Measure for Geometrically Finite Groups Acting on Complex or Quaternionic Hyperbolic Space

The goal of this paper is to provide a tool, the Global Measure Formula, that will facilitate the study of the limit set of discrete geometrically finite groups of isometries of the rank one symmetric spaces. We consider the shadow of a ball from a fixed reference point onto the boundary, and prove a formula that describes the measure of the shadow in terms of the center of the shadowed ball, generalizing a result from real hyperbolic geometry.     

Nonexistence of Invariant Distributions Supported on the Limit Set

We obtain sufficient conditions excluding the existence of nontrivial distribution sections of bundles over the boundary of symmetric spaces of negative curvature which are invariant with respect to a geometrically finite group of isometries and are supported on the limit set in a strong sense.     

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.     

Infinite index subgroups and finiteness properties of intersections of geometrically finite groups

We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup. We produce several examples of such intersections of geometrically finite groups including finitely generated but not finitely presented discrete subgroups.     

Asymptotic Geometry and Growth of Conjugacy Classes of Nonpositively Curved Manifolds

Let X be a Hadamard manifold and Γ⊂Isom(X) a discrete group of isometries which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the limit set of Γ in order to derive a new asymptotic estimate for the growth rate of closed geodesics in not necessarily compact or finite volume manifolds. Mathematics Subject Classifications (2000): 20E45, 53C22, 37F35     

The role of restriction theorems in harmonic analysis on harmonic NA groups

We shall obtain inequalities for Fourier transform via moduli of continuity on NA groups. These results in particular settle the conjecture posed in a recent paper by W.O. Bray and M. Pinsky in the context of noncompact rank one symmetric spaces. These problems naturally demand versions of Fourier restriction theorem on these spaces which we shall prove. We shall also elaborate on the connection between the restriction theorem and the Kunze-Stein phenomena on NA groups. For noncompact Riemannian symmetric spaces of rank one analogues of all the results follow the same way.     

Symmetric spaces of higher rank do not admit differentiable compactifications

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank 1 spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.     

Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

单位球上加权Begrman空间和Hardy空间上的保范复合算子

作者证明了复合算子C_φ是单位球上加权Begrman空间上的保范算子当且仅当φ是旋转变换.对单位球上的Hardy空间上的保范复合算子仅得到部分结果.     

Limits of cubes

The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the “Nim field” (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations.     

Isometric actions on pseudo-Riemannian nilmanifolds

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo- $H$ -type Lie groups.     

Ford fundamental domains in symmetric spaces of rank one

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and domains are essentially subsumed by our work.     

Isometries of ergodic Hardy spaces

The isometries between ergodic Hardy spaces are identified. As a consequence it is shown that the isometry class of an ergodic Hardy space associated with an ergodic, measure preserving flow constitutes an essentially complete set of conjugacy invariants for the flow. This research was supported in part by a grant from the National Science Foundation.     

Isometries of Weighted Spaces of Harmonic Functions

We investigate the isometries between weighted spaces of harmonic functions. We show that, under some mild conditions, every isometry is a composition operator. Our research shows that the structure of isometries of weighted spaces of harmonic functions is, in general, simpler than that observed for weighted spaces of holomorphic functions. Supported by MEC and FEDER Project MTM 2005-08210.     

The space of left-invariant metrics on a Lie group up to isometry and scaling

We study the spaces of left-invariant Riemannian metrics on a Lie group up to isometry, and up to isometry and scaling. In this paper, we see that such spaces can be identified with the orbit spaces of certain isometric actions on noncompact symmetric spaces. We also study some Lie groups whose spaces of left-invariant metrics up to isometry and scaling are small.     

Marked length rigidity of rank one symmetric spaces and their product

In this paper we show that if two Zariski dense representations, from a group G into Iso(X) where X is rank one symmetric space, have the proportional marked length spectrum, then they are conjugate. As a generalization we show that a Zariski dense representation into the isometry group of the product of rank one symmetric spaces is determined by the marked cross ratio.     

A. D. Alexandrov’s problem for non-positively curved spaces in the sense of Busemann

This paper is the last of a series devoted to the solution of Alexandrov’s problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that isometries of a geodesically complete connected at infinity proper Busemann space X are characterized as follows: If a bijection f: X → X and its inverse f ⁻¹ preserve distance 1, then f is an isometry.