7 research outputs found
Length minima for an infinite family of filling closed curves on a one-holed torus
We explicitly find the minima as well as the minimum points of the geodesic
length functions for the family of filling (hence non-simple) closed curves,
(), on a complete one-holed hyperbolic torus in its relative
Teichm\"uller space, where are simple closed curves on the one-holed
torus which intersect exactly once transversely. This provides concrete
examples for the problem to minimize the geodesic length of a fixed filling
closed curve on a complete hyperbolic surface of finite type in its relative
Teichm\"uller space.Comment: 10 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