17 research outputs found
A length comparison theorem for geodesic currents
We work with the space of geodesic currents on a closed
surface of negative Euler characteristic. By prior work of the author with
Sebastian Hensel, each filling geodesic current has a unique
length-minimizing metric in Teichm\"uller space. In this paper, we show
that, on so-called thick components of , the geometries of and are
comparable, up to a scalar depending only on and the topology of . We
also characterize thick components of the projection using only the length
function of .Comment: 46 pages, 29 figures. Added results on identifying short curves,
thick subsurfaces of projectio
An extension of the Thurston metric to projective filling currents
We study the geometry of the space of projectivized filling geodesic currents
. Bonahon showed that Teichm\"uller space,
embeds into . We extend the
symmetrized Thurston metric from to the entire (projectivized)
space of filling currents, and we show that is isometrically
embedded into the bigger space. Moreover, we show that there is no
quasi-isometric projection back down to . Lastly, we study the
geometry of a length-minimizing projection from to defined previously by Hensel and the author.Comment: 16 pages, 3 figure
Enumerative properties of triangulations of spherical bundles over S^1
We give a complete characterization of all possible pairs (v,e), where v is
the number of vertices and e is the number of edges, of any simplicial
triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's
triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1
are unique among all triangulations of (n-1)-dimensional homology manifolds
with first Betti number nonzero, vanishing second Betti number, and 2n+1
vertices.Comment: To appear in European J. of Combinatorics. Many typos fixe
Counting geodesics on expander surfaces
We study properties of typical closed geodesics on expander surfaces of high
genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the
Laplacian. Under an additional systole lower bound assumption, we show almost
every geodesic of length much greater than is non-simple. And
we prove almost every closed geodesic of length much greater than is filling, i.e. each component of the complement of the geodesic is a
topological disc. Our results apply to Weil-Petersson random surfaces, random
covers of a fixed surface, and Brooks-Makover random surfaces, since these
models are known to have uniform spectral gap asymptotically almost surely.
Our proof technique involves adapting Margulis' counting strategy to work at
low length scales.Comment: 55 pages, 7 figures. Minor modification
Coarse density of subsets of
Let be the moduli space of genus Riemann surfaces. We
show that an algebraic subvariety of is coarsely dense with
respect to the Teichm\"uller metric (or Thurston metric) if and only if it is
all of . We apply this to projections of
-orbit closures in the space of abelian
differentials. Moreover, we determine which strata of abelian differentials
have coarsely dense projection to .Comment: 12 pages, 1 figur
Building hyperbolic metrics suited to closed curves and applications to lifting simply
Let be an essential closed curve with at most self-intersections
on a surface with negative Euler characteristic. In this paper,
we construct a hyperbolic metric for which has length at most
, where is a constant depending only on the topology of
. Moreover, the injectivity radius of is at least
. This yields linear upper bounds in terms of self-intersection
number on the minimum degree of a cover to which lifts as a simple
closed curve (i.e. lifts simply). We also show that if is a closed
curve with length at most on a cusped hyperbolic surface ,
then there exists a cover of of degree at most to which lifts simply, for depending only on the topology
of .Comment: 18 pages, 7 figures. Comments welcome
Simple vs non-simple loops on random regular graphs
In this note we solve the ``birthday problem'' for loops on random regular
graphs. Namely, for fixed , we prove that on a random -regular graph
with vertices, as approaches infinity, with high probability:
(i) almost all primitive non-backtracking loops of length
are simple, i.e. do not self-intersect,
(ii) almost all primitive non-backtracking loops of length
self-intersect.Comment: 20 Pages, 1 Figur
Local geometry of random geodesics on negatively curved surfaces
It is shown that the tessellation of a compact, negatively curved surface
induced by a typical long geodesic segment, when properly scaled, looks locally
like a Poisson line process. This implies that the global statistics of the
tessellation -- for instance, the fraction of triangles -- approach those of
the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces
with possibly variable negative curvatur