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 \geq 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.