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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
Abstract

Let S be an infinite-type surface and p 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 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.

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