An upper bound for the crossing number of augmented cubes

A {\it good drawing} of a graph $G$ is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a good drawing of $G$ in the plane. The {\it $n$-dimensional augmented cube} $AQ_n$, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of $AQ_n$ less than $26/324^{n}-(2n^2+7/2n-6)2^{n-2}$.Comment: 39 page

The cyclic cutwidth of mesh cubes

This project\u27s purpose was to understand the workings of a new theorem introduced in a professional paper on the cutwidth of meshes and then use this knowledge to apply it to the search for the cyclic cutwidth of the n-cube

Nice labeling problem for event structures: a counterexample

In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ admits a labeling with at most $f(n)$ labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barth\'elemy and Constantin in 1993

On the quasi-isometric rigidity of graphs of surface groups

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly commensurable with $\Gamma$.Comment: 54 pages, 10 figures, comments welcom

Weak hyperbolicity of cube complexes and quasi-arboreal groups

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of $\Gamma$.Comment: Corrections in Sections 2 and 4. Simplification in Section

Hierarchically hyperbolic spaces I: curve complexes for cubical groups

In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur--Minsky-style distance formula. We then define a \emph{hierarchically hyperbolic space}; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichm\"{u}ller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi, Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the referee's comment

Electronic structure of spinel-type LiV_2O_4

The band structure of the cubic spinel compound LiV_2O_4, which has been reported recently to show heavy Fermion behavior, has been calculated within the local-density approximation using a full-potential version of the linear augmented-plane-wave method. The results show that partially-filled V 3d bands are located about 1.9 eV above the O 2p bands and the V 3d bands are split into a lower partially-filled t_{2g} complex and an upper unoccupied e_{g} manifold. The fact that the conduction electrons originate solely from the t_{2g} bands suggests that the mechanism for the mass enhancement in this system is different from that in the 4f heavy Fermion systems, where these effects are attributed to the hybridization between the localized 4f levels and itinerant spd bands.Comment: 5 pages, revte