Ádám Tímár: A nonamenable `factor' of a Euclidean space

Speaker: Ádám Tímár

Title: A nonamenable `factor' of a Euclidean space

Abstract: Answering a question of Benjamini, we show an isometry-invariant partition of the euclidean space $\R^3$ into indistinguishable connected pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. (Here indistinguishability means that if A is a measurable collection of open subsets of $\R^3$ containing the origin and closed under isometries that keep the origin within the set, either every class of the partition is in A or none.) EEE The existence of such an invariant decomposition, from amenable to nonamenable, is rather unexpected, e.g. there is no way to define a 3-regular infinite tree on the vertices of $\Z^d$ (not necessarily as a subgraph) with an invariant distribution. EEE On the way, it is proved that any finitely generated amenable Cayley graph can be represented in $\R^3$ as an invariant collection of domains, a 3-dimensional analogue of planar maps.