Large-scale geometry of big mapping class groups

Abstract

We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non-locally-compact groups. We give a complete classification of those surfaces whose mapping class groups have local coarse boundedness (the analog of local compactness). When the end space of the surface is countable or tame, we also give a classification of those surfaces where there exists a coarsely bounded generating set (the analog of finite or compact generation, giving the group a well-defined quasi-isometry type) and those surfaces with mapping class groups of bounded diameter (the analog of compactness). We also show several relationships between the topology of a surface and the geometry of its mapping class groups. For instance, we show that nondisplaceable subsurfaces are responsible for nontrivial geometry and can be used to produce unbounded length functions on mapping class groups using a version of subsurface projection; while self-similarity of the space of ends of a surface is responsible for boundedness of the mapping class group.