  We prove that the mapping class group of the one-holed Cantor tree surface is
acyclic. This in turn determines the homology of the mapping class group of the
once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in
particular answering a recent question of Calegari and Chen. We in fact prove
these results for a general class of infinite-type surfaces called binary tree
surfaces. To prove our results we use two main ingredients: one is a
modification of an argument of Mather related to the notion of dissipated
groups; the other is a general homological stability result for mapping class
groups of infinite-type surfaces.
