Non‐amenability of mapping class groups of infinite‐type surfaces and graphs

Abstract This paper completely determines the non‐amenability of the mapping class groups of infinite‐type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non‐amenable stabiliser of a point at infinity of a coarsely bounded generated hyperbolic Polish group, and exhibits a class of mapping class groups of trees or rank‐one graphs that are amenable.