Infinite-type loxodromic isometries of the relative arc graph

Abbott, Carolyn¹ ; Miller, Nicholas² ; Patel, Priyam³

¹Department of Mathematics, Brandeis University, Waltham, MA, United States
²Department of Mathematics, University of Oklahoma, Norman, OK, United States
³Department of Mathematics, University of Utah, Salt Lake City, UT, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 563-644

Résumé

An infinite-type surface Σ is admissible if it has an isolated puncture p and admits shift maps. This includes all infinite-type surfaces with an isolated puncture outside of two sporadic classes. Given such a surface, we construct an infinite family of intrinsically infinite-type mapping classes that act loxodromically on the relative arc graph 𝒜(Σ,p). J Bavard produced such an element for the plane minus a Cantor set, and our result gives the first examples of such mapping classes for all other admissible surfaces. The elements we construct are the composition of three shift maps on Σ, and we give an alternative characterization of these elements as a composition of a pseudo-Anosov on a finite-type subsurface of Σ and a standard shift map. We then explicitly find their limit points on the boundary of 𝒜(Σ,p) and their limiting geodesic laminations. Finally, we show that these infinite-type elements can be used to prove that Map ⁡ (Σ,p) has an infinite-dimensional space of quasimorphisms.

Détail

DOI : 10.2140/agt.2025.25.563

Keywords: big mapping class group, loxodromic, relative arc graph, delta-hyperbolic

Bibliographie

[1] A I O Mathematics, Surfaces of infinite type, problem lists (2019)

[2] J Aramayona, A Fossas, H Parlier, Arc and curve graphs for infinite-type surfaces, Proc. Amer. Math. Soc. 145 (2017) 4995 | DOI

[3] J Aramayona, P Patel, N G Vlamis, The first integral cohomology of pure mapping class groups, Int. Math. Res. Not. 2020 (2020) 8973 | DOI

[4] J Aramayona, N G Vlamis, Big mapping class groups: an overview, from: "In the tradition of Thurston : geometry and topology" (editors K Ohshika, A Papadopoulos), Springer (2020) 459 | DOI

[5] J Bavard, Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire, Geom. Topol. 20 (2016) 491 | DOI

[6] J Bavard, A Genevois, Big mapping class groups are not acylindrically hyperbolic, Math. Slovaca 68 (2018) 71 | DOI

[7] J Bavard, A Walker, The Gromov boundary of the ray graph, Trans. Amer. Math. Soc. 370 (2018) 7647 | DOI

[8] J Bavard, A Walker, Two simultaneous actions of big mapping class groups, Trans. Amer. Math. Soc. 376 (2023) 7603 | DOI

[9] M Bestvina, K Bromberg, K Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 1 | DOI

[10] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69 | DOI

[11] F Bonahon, Closed curves on surfaces, preprint (2000)

[12] D Calegari, Big mapping class groups and dynamics, blog post (2009)

[13] J Cantwell, L Conlon, S R Fenley, Endperiodic automorphisms of surfaces and foliations, Ergodic Theory Dynam. Systems 41 (2021) 66 | DOI

[14] F Fournier-Facio, Y Lodha, M C B Zaremsky, Braided Thompson groups with and without quasimorphisms, Algebr. Geom. Topol. 24 (2024) 1601 | DOI

[15] S Hensel, P Przytycki, R C H Webb, 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. 17 (2015) 755 | DOI

[16] W P Hooper, The invariant measures of some infinite interval exchange maps, Geom. Topol. 19 (2015) 1895 | DOI

[17] B Von Kerékjártó, Vorlesungen über Topologie, I : Flächentoplogie, Springer (1923)

[18] J Lanier, M Loving, Centers of subgroups of big mapping class groups and the Tits alternative, Glas. Mat. Ser. III 55(75) (2020) 85 | DOI

[19] W B R Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. 76 (1962) 531 | DOI

[20] I Morales, F Valdez, Loxodromic elements in big mapping class groups via the Hooper–Thurston–Veech construction, Algebr. Geom. Topol. 22 (2022) 3809 | DOI

[21] J Nielsen, Surface transformation classes of algebraically finite type, 2, Munksgaard (1944) 89

[22] P Patel, N G Vlamis, Algebraic and topological properties of big mapping class groups, Algebr. Geom. Topol. 18 (2018) 4109 | DOI

[23] R C Penner, J L Harer, Combinatorics of train tracks, 125, Princeton Univ. Press (1992) | DOI

[24] A J Rasmussen, WWPD elements of big mapping class groups, Groups Geom. Dyn. 15 (2021) 825 | DOI

[25] I Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963) 259 | DOI

[26] D Šarić, Train tracks and measured laminations on infinite surfaces, Trans. Amer. Math. Soc. 374 (2021) 8903 | DOI

[27] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417 | DOI

[28] Y Verberne, A construction of pseudo-Anosov homeomorphisms using positive twists, Algebr. Geom. Topol. 23 (2023) 1601 | DOI