Graphs of curves and arcs quasi-isometric to big mapping class groups

2020/06/26 by Anschel Schaffer-Cohen, Schaffer-Cohen, Anschel · 6 citations
Mathematics · #57K20 #Analytic and geometric function theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2006.14760

openalex publication_date 2020/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Following the work of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically nontrivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first example of a mapping class group known to be non-elementary hyperbolic.

Cited by

• Coarse geometry of homeomorphism groups: Classifying countable Stone spaces
• Dense Conjugacy Classes and Stability of Locally Finite Graphs
• The number of ends of big mapping class groups
• Strongly bounded generation in transformation groups
• Geometric models and asymptotic dimension for infinite-type surface mapping class groups
• Non-amenability of mapping class groups of infinite-type surfaces and graphs

Related

• Geometry and rigidity of mapping class groups
• Automorphisms of the loop and arc graph of an infinite-type surface
• On a lemma of Milnor and Schwarz, après Rosendal
• Mapping class groups are quasicubical
• The number of ends of big mapping class groups