In a beautiful paper, Sleator, Tarjan and Thurston solved the problem of maximum rotation distance of two binary trees. Equivalently they solved the problem of rotation distance of triangulations on the disk. We extend their results to rotation distance of triangulations of other planar surfaces. We give upper and lower bounds for this problem. Equivalently, by duality, one can interpret our re...

We begin with some covering theory for complexes of groups. For example, we show that a covering of developable complexes of groups induces a monomorphism of fundamental groups, and an equivariant isometry of universal covers. As an application, we then investigate the asymptotics of the number of “overlattices” of a cocompact lattice Γ in Aut(K), where K is a locally finite polyhedral complex....

This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, implies many striking results about the geometry and analysis of the manifold or i...

A family of sets F is locally k-wide if and only if the width (as a poset ordered by inclusion) of F x = fU 2 F j x 2 U g is at most k for every x. The directed covering graph of a locally 1-wide family of sets is a forest of rooted trees. It is shown that if F is a locally k-wide family of subsets of f1; : : : ; ng, then jFj (2k) k?1 n. The proof involves a counting argument based on families ...

The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the dimension of H1(M ;Z2) is at most 7, and that it is at most 6 unless M is “strange.” To say that a closed, orientable 3-manifold M , for which H1(M ;Z2) has dimension 7, is strange means that the Z2-vector space H1(M ;Z2) has a 2-dimensional subspace X such that for every...

Let X be a geodesic space. We say that X is geodesically complete if every geodesic segment β : [0, a] → X from β(0) to β(a) can be extended to a geodesic ray α : [0,∞) → X, (i.e. β(t) = α(t), for 0 ≤ t ≤ a). If X is a compact npc space (“npc” means: “non-positively curved”) then it is almost geodesically complete, see [10]. (X, with metric d, is almost geodesically complete if its universal co...

This paper contains a loose collection of remarks on F1-schemes. Etale morphisms and universal coverings are introduced. The relation to toric varieties, at least for integral schemes, is clarified.

We study a class of geometric stabbing/covering problems for sets of line segments, rays, and lines in the plane. While we demonstrate that the problems on sets of horizontal/vertical line segments are NP-complete, we show that versions involving (parallel) rays or lines are polynomially solvable.

Let k3(n) denote the minimal cardinality of a ternary code of length n and covering radius one. In this paper we show k3(7) ≥ 156 and k3(8) ≥ 402 improving on the best previously known bounds k3(7) ≥ 153 and k3(8) ≥ 398. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in k...