One-ended spanning trees in amenable unimodular graphs

Abstract: We prove that every amenable one-ended Cayley graph has an invariant spanning tree of one end. More generally, for any 1-ended amenable unimodular random graph we construct a factor of iid percolation (jointly unimodular subgraph) that is almost surely a spanning tree of one end. In [2] and [1] similar claims were proved, but the resulting spanning tree had 1 or 2 ends, and one had no control of which of these two options would be the case.