Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces

2023/08/31 by Hill, Thomas · 2 citations
#57K20 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2309.00124

Abstract

The work of Mann and Rafi gives a classification surfaces Σ when \textrmMap(Σ) is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann--Rafi, we prove that \textrmPMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface, and locally CB or CB generated if and only if Σ has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures.

Cited by

• The number of ends of big mapping class groups
• Geometric models and asymptotic dimension for infinite-type surface mapping class groups

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