One-ended spanning trees in amenable unimodular graphs

TL;DR

This paper proves that all amenable one-ended Cayley graphs and unimodular random graphs admit invariant spanning trees with exactly one end, improving control over the number of ends compared to previous results.

Contribution

It introduces a method to construct invariant spanning trees with a single end in amenable unimodular graphs, advancing beyond prior work with less precise end control.

Findings

Every amenable one-ended Cayley graph has an invariant one-ended spanning tree.

Constructs a factor of iid percolation yielding a one-ended spanning tree in unimodular random graphs.

Improves previous results by ensuring the spanning tree has exactly one end.

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.