Minimal pseudo-Anosov translation lengths on the complex of curves

TL;DR

This paper provides bounds on the minimal asymptotic translation lengths of pseudo-Anosov homeomorphisms on the curve complex of orientable surfaces, showing they scale inversely with the square of the genus.

Contribution

It establishes explicit bounds on minimal pseudo-Anosov translation lengths in relation to the surface genus, refining understanding of their asymptotic behavior.

Findings

Minimal translation length scales as 1/g^2 for genus g surfaces.

Bounds are established with explicit constants a_1 and a_2.

Results apply to orientable closed surfaces with genus ≥ 2.

Abstract

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1 < a_2$ such that the minimal translation length is bounded below and above by $a_1/ g^2$ and $a_2/g^2$.