15,156 research outputs found
The shadow of a Thurston geodesic to the curve graph
We study the geometry of the Thurston metric on Teichmuller space by
examining its geodesics and comparing them to Teichmuller geodesics. We show
that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to
the curve graph is a reparametrized quasi-geodesic. However, we show that the
set of short curves along the two geodesics are not identical.Comment: 34 pages, 10 figures, minor revisions, final version appears in
Journal of Topolog
The asymptotic dimension of a curve graph is finite
We find an upper bound for the asymptotic dimension of a hyperbolic metric
space with a set of geodesics satisfying a certain boundedness condition
studied by Bowditch. The primary example is a collection of tight geodesics on
the curve graph of a compact orientable surface. We use this to conclude that a
curve graph has finite asymptotic dimension. It follows then that a curve graph
has property . We also compute the asymptotic dimension of mapping class
groups of orientable surfaces with genus .Comment: 19 pages. Made some minor revisions. The section on mapping class
groups has been rewritten; in particular we compute the asdim of Mod(S) where
S has genus at most 2. The last section on open questions has been modified
to reflect recent developments. References have been update
Generalized chordality, vertex separators and hyperbolicity on graphs
Let be a graph with the usual shortest-path metric. A graph is
-hyperbolic if for every geodesic triangle , any side of is
contained in a -neighborhood of the union of the other two sides. A
graph is chordal if every induced cycle has at most three edges. A vertex
separator set in a graph is a set of vertices that disconnects two vertices. In
this paper we study the relation between vertex separator sets, some chordality
properties which are natural generalizations of being chordal and the
hyperbolicity of the graph. We also give a characterization of being
quasi-isometric to a tree in terms of chordality and prove that this condition
also characterizes being hyperbolic, when restricted to triangles, and having
stable geodesics, when restricted to bigons.Comment: 16 pages, 3 figure
Limiting geodesics for first-passage percolation on subsets of
It is an open problem to show that in two-dimensional first-passage
percolation, the sequence of finite geodesics from any point to has a
limit in . In this paper, we consider this question for first-passage
percolation on a wide class of subgraphs of : those whose vertex
set is infinite and connected with an infinite connected complement. This
includes, for instance, slit planes, half-planes and sectors. Writing for
the sequence of boundary vertices, we show that the sequence of geodesics from
any point to has an almost sure limit assuming only existence of finite
geodesics. For all passage-time configurations, we show existence of a limiting
Busemann function. Specializing to the case of the half-plane, we prove that
the limiting geodesic graph has one topological end; that is, all its infinite
geodesics coalesce, and there are no backward infinite paths. To do this, we
prove in the Appendix existence of geodesics for all product measures in our
domains and remove the moment assumption of the Wehr-Woo theorem on absence of
bigeodesics in the half-plane.Comment: Published in at http://dx.doi.org/10.1214/13-AAP999 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- âŠ