One-ended spanning subforests and treeability of groups
Abstract
We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. This provides the first examples of one-ended nonamenable groups which are measure strongly treeable. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests.
Keywords
Cite
@article{arxiv.2104.07431,
title = {One-ended spanning subforests and treeability of groups},
author = {Clinton T. Conley and Damien Gaboriau and Andrew S. Marks and Robin D. Tucker-Drob},
journal= {arXiv preprint arXiv:2104.07431},
year = {2026}
}
Comments
Various typos fixed. Small changes according to referees suggestions. 47 pages, 2 x 2 figures