Infinite-type loxodromic isometries of the relative arc graph

Abbott, Carolyn; Miller, Nicholas; Patel, Priyam

doi:10.2140/agt.2025.25.563

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Infinite-type loxodromic isometries of the relative arc graph

Carolyn Abbott, Nicholas Miller and Priyam Patel

Algebraic & Geometric Topology 25 (2025) 563–644

DOI: 10.2140/agt.2025.25.563

Abstract

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.

Keywords

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

Mathematical Subject Classification

Primary: 20F65, 57K20

References

Publication

Received: 9 August 2023

Accepted: 13 November 2023

Published: 24 March 2025

Authors

Carolyn Abbott
Department of Mathematics Brandeis University Waltham, MA United States

Nicholas Miller
Department of Mathematics University of Oklahoma Norman, OK United States

Priyam Patel
Department of Mathematics University of Utah Salt Lake City, UT United States

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Volume 25, issue 1 (2025)