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 selfsimilarity of the space of ends of a surface is responsible for boundedness of the mapping class group.

Two results on end spaces of infinite type surfaces

“…This appeared as Question 1.4 in [4], and Remark 6.2 in [5]. Here we answer the question, showing the strict converse (without extra assumptions) to [6,Prop. 4.8] is false:…”

“…This notion has turned out to be a useful one and it been appeared many times in the literature, see for instance [1,2,3,7]. A partial converse to this statement was proved in [6,Prop. 4.8].…”

“…We say x, y ∈ E are non-comparable if neither x y nor y x holds. See [6] for more details and discussion.…”

“…The paper [6] introduced the notion of self-similar end spaces for infinite type surfaces, and proved that a self-similar end space necessarily contains a unique maximal type of end, with the set of ends of this type either a singleton or a Cantor set. See Section 2 for the definition of the partial order on ends types.…”

“…To prove this, we first provide details on a local construction of non-comparable points (an idea sketched loosely in [6]), and then use this to build examples first in the case where the maximal end is a singleton, followed by the Cantor set case.…”