Every Anosov map of the torus has a bi-infinite geodesic axis in the Farey graph, making its stable translation length a computable positive integer.
Polynomial-time algorithms for the curve graph
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abstract
We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For example, we can use these geodesics to compute the asymptotic translation length, Nielsen--Thurston type, and canonical curve system of a mapping class in polynomial time in its word length.
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On Translation Lengths of Anosov Maps on Curve Graph of Torus
Every Anosov map of the torus has a bi-infinite geodesic axis in the Farey graph, making its stable translation length a computable positive integer.