Super efficiency of efficient geodesics in the complex of curves

Abstract

We show that efficient geodesics have the strong property of "super efficiency". For any two vertices, $v,w\in \mathcal{C}(S_g)$, in the complex of curves of a closed oriented surface of genus $g\ge 2$, and any efficient geodesic, $v=v_1,\cdots ,v_{\text{d}}=w$, it was previously established by Birman, Margalit and the second author (see arXiv:1408.4133) that there is an explicitly computable list of at most ${\text{d}}^{(6g-6)}$ candidates for the $v_1$ vertex. In this note we establish a bound for this computable list that is independent of $\text{d}$-distance and only dependent on genus -- the super efficiency property. The proof relies on a new intersection growth inequality between intersection number of curves and their distance in the complex of curves, together with a thorough analysis of the dot graph associated with the intersection sequence.

For Version 4, improved introduction in explaining new idea utilizing intersection growth function. An additional section at the end discussing how super efficient geodesics can be utilized in a distance algorithm

References in corpus (1)

• Polynomial-time algorithms for the curve graph