Loxodromic elements in big mapping class groups via the Hooper–Thurston–Veech construction

Morales, Israel; Valdez, Ferrán

doi:10.2140/agt.2022.22.3809

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Loxodromic elements in big mapping class groups via the Hooper–Thurston–Veech construction

Israel Morales and Ferrán Valdez

Algebraic & Geometric Topology 22 (2022) 3809–3854

DOI: 10.2140/agt.2022.22.3809

Abstract

Let $S$ be an infinite-type surface and $p \in S$ . We show that the Thurston–Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $Mod (S; p)$ on the loop graph $L (S; p)$ that do not leave any finite-type subsurface $S^{'} \subset S$ invariant. Moreover, in the language of Bavard and Walker, the Thurston–Veech construction produces loxodromic elements of any weight. As a consequence of Bavard and Walker’s work, any subgroup of $Mod (S; p)$ containing two “Thurston–Veech loxodromics” of different weight has an infinite-dimensional space of nontrivial quasimorphisms.

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Keywords

big mapping class groups, loxodromic actions, pseudo-Anosov

Mathematical Subject Classification

Primary: 20F65, 37E30, 57M60

References

Publication

Received: 8 May 2020

Revised: 18 August 2021

Accepted: 13 September 2021

Published: 14 March 2023

Authors

Israel Morales
Instituto de Matemáticas Unidad Oaxaca Universidad Nacional Autonóma de México Oaxaca Mexico

Ferrán Valdez
Centro de Ciencias Matemáticas, Campus Morelia Universidad Nacional Autonóma de México Morelia Mexico

Volume 22, issue 8 (2022)