11 research outputs found
Filling of closed Surfaces
Let denote a closed oriented surface of genus . A set of simple
closed curves is called a filling of if its complement is a disjoint
union of discs. The mapping class group of genus acts on
the set of fillings of . The union of the curves in a filling forms a
graph on the surface which is a so-called decorated fat graph. It is a fact
that two fillings of are in the same -orbit if and only
if the corresponding fat graphs are isomorphic. We prove that any filling of
whose complement is a single disc (i.e., a so-called minimal filling) has
either three or four closed curves and in each of these two cases, there is a
unique such filling up to the action of .
We provide a constructive proof to show that the minimum number of discs in
the complement of a filling pair of is two. Finally, given positive
integers and with , we construct a filling pair of
such that the complement is a union of topological discs.Comment: 15 Pages, 11 Figures, To appear in J. Topol. Ana
Embedding of metric graphs on hyperbolic surfaces
An embedding of a metric graph on a closed hyperbolic surface is
\emph{essential}, if each complementary region has a negative Euler
characteristic. We show, by construction, that given any metric graph, its
metric can be rescaled so that it admits an essential and isometric embedding
on a closed hyperbolic surface. The essential genus of is the
lowest genus of a surface on which such an embedding is possible. In the next
result, we establish a formula to compute . Furthermore, we show that
for every integer , admits such an embedding (possibly
after a rescaling of ) on a surface of genus .
Next, we study minimal embeddings where each complementary region has Euler
characteristic . The maximum essential genus of is
the largest genus of a surface on which the graph is minimally embedded.
Finally, we describe a method explicitly for an essential embedding of , where and are realized.Comment: Revised version, 11 pages, 3 figure
Filling with separating curves
A pair of simple closed curves on a closed and orientable
surface of genus is called a filling pair if the complement is a
disjoint union of topological disks. If is separating, then we call it
as separating filling pair. In this article, we find a necessary and sufficient
condition for the existence of a separating filling pair on with exactly
two complementary disks. We study the combinatorics of the action of the
mapping class group \M on the set of such filling pairs. Furthermore, we
construct a Morse function on the moduli space
which, for a given hyperbolic surface , outputs the length of shortest such
filling pair with respect to the metric in . We show that the cardinality of
the set of global minima of the function is the same as the
number of \M-orbits of such filling pairs.Comment: 30 Pages, 16 Figures, Theorem 1.3, Subsection 4.1, Lemma 7.8 are
added to the previous versio