Generic leaves

A remarkable theorem of E. Ghys asserts that, for any harmonic measure on a compact, foliated metric space, -almost every leaf has 0, 1, 2 or a Cantor set of ends. In this paper, analogous results are proven for topologically almost all (i.e., residual families of) leaves. More precisely, if some leaf is totally recurrent, a residual family of leaves is totally recurrent with 1, 2 or a Cantor set of ends. A "local" version of this theorem asserts that, in general, topologically almost all leaves have 0, 1, 2 or a Cantor set of dense ends. Received: October 1, 1997     

Stability of symplectic leaves

We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stability results for foliations and group actions.     

Category and compact leaves

We prove that if F is a C¹-foliation of a compact manifold M with finite transverse saturated LS category, , then F has a compact leaf. In contrast, we show that if F is expansive on some non-trivial minimal set of F, then . Examples of foliations are given to illustrate the main results of the paper.     

Bounded geometry and leaves

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.     

Compact leaves of structurally stable foliations

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspension foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary compact manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in the sense of Ehresmann and Reeb.     

Bounds on leaves of one-dimensional foliations

Let X be a variety over an algebraically closed field, a onedimensional singular foliation, and a projective leaf of . We prove that

Foliations with leaves of nonpositive curvature

We answer a question of Gromov ([G2]) in the codimension 1 case: ifF is a codimension 1 foliation of a compact manifoldM with leaves of negative curvature, thenπ ₁(M) has exponential growth. We also prove a result analogous to Zimmer’s ([Z2]): ifF is a codimension 1 foliation on a compact manifold with leaves of nonpositive curvature, and ifπ ₁(M) has subexponential growth, then almost every leaf is flat. We give a foliated version of the Hopf theorem on surfaces without conjugate points. Partially supported by NSF Grant #DMS 9403870.     

Spanning trees with many leaves

We show that if G is a simple connected graph with and , then G has a spanning tree with > t leaves, and this is best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 189–197, 2001     

Symplectic leaves and deformation quantization

In this paper, we show that for any classical simple compact Poisson Lie group , there is no quantization of using the quantum group , which is both group-preserving and symplectic leaf-preserving.

     

Light leaves and Lusztig's conjecture

We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the following property of F  : if the image p=F(l)$p=F(l)$ of some light leaf l under F is bigger than the Coxeter number of the corresponding Weyl group, then there is a counterexample to Lusztig's conjecture in characteristic p. We also introduce the “double leaves basis” which is an improvement of the light leaves basis that has already found interesting applications. In particular it forms a cellular basis of Soergel bimodules that allows us to produce an algorithm to find “the bad primes” for Lusztig's conjecture.     

Spanning trees with many leaves

Let a maximal chain of vertices of degree 2 in a graph G consist of k > 0 vertices. We prove that G has a spanning tree with more than \fracv(G)2k + 4 \frac{{v(G)}}{{2k + 4}} leaves (where υ(G) is the number of vertices of the graph G). We present an infinite series of examples showing that the constant \frac12k + 4 \frac{1}{{2k + 4}} cannot be enlarged. Bibliography: 7 titles.     

Foliated control with hyperbolic leaves

The main result is a control theorem for the space of stable pseudo-isotopies on E with control near the leaves of F in M, where : E M is a fiber bundle over the compact closed Riemannian manifold M having a compact manifold for fiber and F is a smooth foliation of M. The proof of this foliated control result combines the (unfoliated) control results of [14] with the long and thin cell structures and the asymptotic transfer of [6] and [7].Both authors were supported in part by the NSF.     

Noncompact leaves of foliations of Morse forms

In this paper foliations determined by Morse forms on compact manifolds are considered. An inequality involving the number of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 862–865, June, 1998.The author wishes to thank Professor A. S. Mishchenko for his interest in this work and stimulating discussions.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00276.     

Quadratic leaves of maximal partial triple systems

Every graph having vertex degrees zero and two satisfying the basic necessary conditions is the leave of a maximal partial triple system, with one exception (C ₄ C ₅). The proof technique is direct, using the method of differences.