99,747 research outputs found

    Statistical hyperbolicity in groups

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    In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (i.e., without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products, for Diestel-Leader graphs and lamplighter groups. In free abelian groups, the word metrics asymptotically approach norms induced by convex polytopes, causing the study of sprawl to reduce to a problem in convex geometry. We present an algorithm that computes sprawl exactly for any generating set, thus quantifying the failure of various presentations of Z^d to be hyperbolic. This leads to a conjecture about the extreme values, with a connection to the classic Mahler conjecture.Comment: 14 pages, 5 figures. This is split off from the paper "The geometry of spheres in free abelian groups.

    Explicit expanders with cutoff phenomena

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    The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the authors showed this for a typical random regular graph. However, no example was known for an explicit (deterministic) family of expanders with this phenomenon. Here we construct a family of cubic expanders where the random walk from a worst case initial position exhibits total-variation cutoff. Variants of this construction give cubic expanders without cutoff, as well as cubic graphs with cutoff at any prescribed time-point.Comment: 17 pages, 2 figure

    Self-avoiding walks and connective constants

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    The connective constant μ(G)\mu(G) of a quasi-transitive graph GG is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on GG from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph GG. ∙\bullet We present upper and lower bounds for μ\mu in terms of the vertex-degree and girth of a transitive graph. ∙\bullet We discuss the question of whether μ≥ϕ\mu\ge\phi for transitive cubic graphs (where ϕ\phi denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). ∙\bullet We present strict inequalities for the connective constants μ(G)\mu(G) of transitive graphs GG, as GG varies. ∙\bullet As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. ∙\bullet We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. ∙\bullet A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. ∙\bullet Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. ∙\bullet The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

    Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

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    We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) 33-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the 33-connected case) to this setting. Precisely, for any cylindric essentially internally 33-connected map GG with nn vertices, we can obtain in linear time a periodic (in xx) straight-line drawing of GG that is crossing-free and internally (weakly) convex, on a regular grid Z/wZ×[0..h]\mathbb{Z}/w\mathbb{Z}\times[0..h], with w≤2nw\leq 2n and h≤n(2d+1)h\leq n(2d+1), where dd is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially 33-connected map GG on the torus (i.e., 33-connected in the periodic representation) with nn vertices, we can compute in linear time a periodic straight-line drawing of GG that is crossing-free and (weakly) convex, on a periodic regular grid Z/wZ×Z/hZ\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}, with w≤2nw\leq 2n and h≤1+2n(c+1)h\leq 1+2n(c+1), where cc is the face-width of GG. Since c≤2nc\leq\sqrt{2n}, the grid area is O(n5/2)O(n^{5/2}).Comment: 37 page
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