Counting Mapping Class group orbits on hyperbolic surfaces
Abstract
Let
S
g,n
be a surface of genus
g
with
n
marked points. Let
X
be a complete hyperbolic metric on
S
g,n
with
n
cusps. Every isotopy class
[γ]
of a closed curve
γ∈
π
1
(
S
g,n
)
contains a unique closed geodesic on
X
.
Let
ℓ
γ
(X)
denote the hyperbolic length of the geodesic representative of
γ
on
X
. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on
S
g,n
.
As an application, one can obtain the asymptotics of the growth of
s
k
X
(L)
, the number of closed curves of length
≤L
on
X
with at most
k
self-intersections. We also discuss properties of random pants decomposition of large length on
X
. Both these results are based on ergodic properties of the earthquake flow on a natural bundle over the moduli space
M
g,n
of hyperbolic surfaces of genus
g
with
n
cusps.