We give an algorithm to compute the stable lengths of pseudo-Anosovs on the curve graph, answering a question of Bowditch. We also give a procedure to compute all invariant tight geodesic axes of pseudo-Anosovs.
Along the way we show that there are constants
1
>
a
1
>
a
2
1>a_1>a_2
such that the minimal upper bound on ‘slices’ of tight geodesics is bounded below and above by
a
1
ξ
(
S
)
a_1^{\xi (S)}
and
a
2
ξ
(
S
)
a_2^{\xi (S)}
, where
ξ
(
S
)
\xi (S)
is the complexity of the surface. As a consequence, we give the first computable bounds on the asymptotic dimension of curve graphs and mapping class groups.
Our techniques involve a generalization of Masur–Minsky’s tight geodesics and a new class of paths on which their tightening procedure works.