Bulletin of the
Korean Mathematical Society

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016



Bull. Korean Math. Soc. 2024; 61(3): 585-595

Online first article May 17, 2024      Printed May 31, 2024


Copyright © The Korean Mathematical Society.

On translation lengths of pseudo-Anosov maps on the curve graph

Hyungryul Baik, Changsub Kim



We show that a pseudo-Anosov map constructed as a product of the large power of Dehn twists of two filling curves always has a geodesic axis on the curve graph of the surface. We also obtain estimates of the stable translation length of a pseudo-Anosov map, when two filling curves are replaced by multicurves. Three main applications of our theorem are the following: (a) determining which word realizes the minimal translation length on the curve graph within a specific class of words, (b) giving a new class of pseudo-Anosov maps optimizing the ratio of stable translation lengths on the curve graph to that on Teichm{\" u}ller space, (c) giving a partial answer of how much power is needed for Dehn twists to generate right-angled Artin subgroup of the mapping class group.

Keywords: Curve graph, pseudo-Anosov map, stable translation length, ratio optimizer, right-angled Artin group

MSC numbers: Primary 57M99, 37E30

Supported by: Both authors were supported by National Research Foundation of Korea(NRF) grant funded by the Korean government(MSIT) (No.~2020R1C1C1A01006912).