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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

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.

big mapping class groups, loxodromic actions, pseudo-Anosov
Mathematical Subject Classification
Primary: 20F65, 37E30, 57M60
Received: 8 May 2020
Revised: 18 August 2021
Accepted: 13 September 2021
Published: 14 March 2023
Israel Morales
Instituto de Matemáticas Unidad Oaxaca
Universidad Nacional Autonóma de México
Ferrán Valdez
Centro de Ciencias Matemáticas, Campus Morelia
Universidad Nacional Autonóma de México