Super efficiency of efficient geodesics in the complex of curves

TL;DR

This paper proves that efficient geodesics in the complex of curves exhibit a super efficiency property, with bounds on initial vertices independent of distance and dependent only on genus, using intersection growth inequalities.

Contribution

It establishes a genus-dependent bound for the initial vertex in efficient geodesics, improving previous distance-dependent bounds, and introduces new intersection inequalities.

Findings

Bound for initial vertex list independent of distance

Intersection growth inequality between curves and geodesic distance

Analysis of dot graphs for intersection sequences

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.