We use random walks to show that the Basilica group is amenable and thus answering an open question of Grigorchuk and Żuk . Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.
Laurent Bartholdi. Bálint Virág. "Amenability via random walks." Duke Math. J. 130 (1) 39 - 56, 01 October 2005. https://doi.org/10.1215/S0012-7094-05-13012-5