Logo CCM

Sistema de Referencias Bibliográficas

Centro de Ciencias Matemáticas UNAM

Usuario: guest
No has iniciado sesión
Type: Article

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

Abstract:

Let Mod(S;p) denote the mapping class group of an orientable infinite-type surface S (i.e., with a not finitely generated fundamental group), with a marked point p fixed by all homeomorphisms and isotopies. The group Mod(S;p) acts by isometries on the loop graph L(S;p) of the surface (of loops based at p), and one of the main objectives of the present paper is the construction and study of loxodromic elements for this action. The authors use the description of the Gromov boundary of L(S;p) in terms of long rays given by J. Bavard and A. K. Walker in [Trans. Amer. Math. Soc. 370 (2018), no. 11, 7647–7678; MR3852444], where also the weight of a loxodromic element is defined. Before, the only known example of a loxodromic element not supported on a compact subsurface was defined by Bavard [Geom. Topol. 20 (2016), no. 1, 491–535; MR3470720], of weight 1. In the present paper, adapting constructions of Hooper, Thurston and Veech for pseudo-Anosov elements to the context of infinite-type surfaces, the authors construct loxodromic elements in Mod(S;p) that do not leave invariant any finite-type subsurface, and this construction produces loxodromic elements of any weight (answering a suggestion of Bavard and Walker; see also a recent preprint of C. R. Abbott, N. Miller and P. Patel ["Infinite-type loxodromic isometries of the relative arc graph'', preprint, arXiv:2109.06106] for the construction of infinite-type loxodromic elements). "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: Group theory and generalizations||Special aspects of finite or finite groups||Geometric group theory
MSC: 20F65 (37E30 57M60)
Journal: Algebraic and Geometric Topology
ISSN: 1472-2739
Year: 2022
Volume: 22
Number: 8
Pages: 3809
Created Created: 2025-05-12 19:05:47
Modified Modified: 2025-05-12 19:06:10
Warn Referencia revisada
Autores Institucionales Asociados a la Referencia: