TITLE: Translation distance in the curve complex
QUESTION [13 upvotes]: Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation distance is 0, 1, 2, 3, many, then I am happy...
I know there are algorithms for computing distances IN the curve complex, but this is not quite the same...

REPLY [7 votes]: Recently, Richard Webb and myself gave a polynomial-time algorithm for computing the asymptotic translation length of a mapping class
$$ \ell(h) = \lim_{n \to \infty} d(x, h^n(x)). $$
This appears as Algorithm 6 of the paper and relies on being able to compute geodesics in the curve complex. Assuming this, the key observation is that a midpoint $c'$ of a curve $c$ and its image under a large enough power of $h$ lies close to the tight axis of $h$. Hence $d(c', h^n(c')) \approx n \ell(h)$. Therefore, since $\ell(h)$ is a rational number with bounded denominator, if we take $n$ large enough then
$$ \ell(h) = \frac{1}{n} d(c', h^n(c')). $$
If you fix a finite generating set $\langle X \rangle = \textrm{Mod}(S)$ then the running time of this algorithm is a polynomial function of the word length of the mapping class $|h|_X$ since this is the running time of our algorithm for computing a geodesic from $c$ to $h(c)$ (Algorithm 4).