129 research outputs found
The homology systole of hyperbolic Riemann surfaces
The main goal of this note is to show that the study of closed hyperbolic
surfaces with maximum length systole is in fact the study of surfaces with
maximum length homological systole. The same result is shown to be true for
once-punctured surfaces, and is shown to fail for surfaces with a large number
of cusps.Comment: 7 pages, 5 figure
Simple closed geodesics and the study of Teichm\"uller spaces
The goal of the chapter is to present certain aspects of the relationship
between the study of simple closed geodesics and Teichm\"uller spaces.Comment: to appear in Handbook of Teichm\"uller theory, vol II
Kissing numbers for surfaces
The so-called {\it kissing number} for hyperbolic surfaces is the maximum
number of homotopically distinct systoles a surface of given genus can
have. These numbers, first studied (and named) by Schmutz Schaller by analogy
with lattice sphere packings, are known to grow, as a function of genus, at
least like g^{\sfrac{4}{3}-\epsilon} for any . The first goal of
this article is to give upper bounds on these numbers; in particular the growth
is shown to be sub-quadratic. In the second part, a construction of (non
hyperbolic) surfaces with roughly g^{\sfrac{3}{2}} systoles is given.Comment: 20 pages, 9 figure
The genus of curve, pants and flip graphs
This article is about the graph genus of certain well studied graphs in
surface theory: the curve, pants and flip graphs. We study both the genus of
these graphs and the genus of their quotients by the mapping class group. The
full graphs, except for in some low complexity cases, all have infinite genus.
The curve graph once quotiented by the mapping class group has the genus of a
complete graph so its genus is well known by a theorem of Ringel and Youngs.
For the other two graphs we are able to identify the precise growth rate of the
graph genus in terms of the genus of the underlying surface. The lower bounds
are shown using probabilistic methods.Comment: 26 pages, 9 figure
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
The maximum number of systoles for genus two Riemann surfaces with abelian differentials
In this article, we provide bounds on systoles associated to a holomorphic
-form on a Riemann surface . In particular, we show that if
has genus two, then, up to homotopy, there are at most systolic loops on
and, moreover, that this bound is realized by a unique translation
surface up to homothety. For general genus and a holomorphic 1-form
with one zero, we provide the optimal upper bound, , on the
number of homotopy classes of systoles. If, in addition, is hyperelliptic,
then we prove that the optimal upper bound is .Comment: 41 page
Systoles and kissing numbers of finite area hyperbolic surfaces
We study the number and the length of systoles on complete finite area
orientable hyperbolic surfaces. In particular, we prove upper bounds on the
number of systoles that a surface can have (the so-called kissing number for
hyperbolic surfaces). Our main result is a bound which only depends on the
topology of the surface and which grows subquadratically in the genus.Comment: A minor mistake and a computation fixed, small changes in the
exposition. 23 pages, 13 figure
Once punctured disks, non-convex polygons, and pointihedra
We explore several families of flip-graphs, all related to polygons or
punctured polygons. In particular, we consider the topological flip-graphs of
once-punctured polygons which, in turn, contain all possible geometric
flip-graphs of polygons with a marked point as embedded sub-graphs. Our main
focus is on the geometric properties of these graphs and how they relate to one
another. In particular, we show that the embeddings between them are strongly
convex (or, said otherwise, totally geodesic). We also find bounds on the
diameters of these graphs, sometimes using the strongly convex embeddings.
Finally, we show how these graphs relate to different polytopes, namely type D
associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure
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