Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper7.abs.htm

Hausdorff dimension in graph matchbox manifolds

In this paper, we study the Hausdorff and the box dimensions of closed invariant subsets of the space of pointed trees, equipped with a pseudogroup action. This pseudogroup dynamical system can be regarded as a generalization of a shift space. We show that the Hausdorff dimension of the space of pointed trees is infinite, and the union of closed invariant subsets with dense orbit and non-equal Hausdorff and box dimensions is dense in the space of pointed trees. We apply our results to the problem of embedding laminations into differentiable foliations of smooth manifolds. To admit such an embedding, a lamination must satisfy at least the following two conditions: first, it must admit a metric and a foliated atlas, such that the generators of the holonomy pseudogroup, associated to the atlas, are bi-Lipschitz maps relative to the metric. Second, it must admit an embedding into a manifold, which is a bi-Lipschitz map. A suspension of the pseudogroup action on the space of pointed graphs gives an example of a lamination where the first condition is satisfied, and the second one is not satisfied, with Hausdorff dimension of the space of pointed trees being the obstruction to the existence of a bi-Lipschitz embedding.Comment: Proof of Theorem 1.1 simplified as compared to the previous version; Sections 5 and 6 contain new result

Reparametrizations of Continuous Paths

A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can be understood via ``stop-maps'' that allow to investigate compositions and factorizations, and we compare it to the distributive lattice of countable subsets of the unit interval. The results obtained are used to analyse the space of traces in a topological space, i.e., the space of continuous paths up to reparametrization equivalence. This space is shown to be homeomorphic to the space of regular paths (without stops) up to increasing reparametrizations. Directed versions of the results are important in directed homotopy theory

Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\mathscr{I}_\lambda^n$ of the rank $\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.Comment: With the participation of the new coauthor - Jimmie Lawson - the manuscript has been substantially revised and expanded. Accordingly, we have also changed the manuscript titl