One-ended spanning subforests and
treeability of groups
Clinton T. Conley, Damien Gaboriau, Andrew S. Marks, Robin D.
Tucker-Drob
Download: pdf
HAL: https://hal.archives-ouvertes.fr/hal-03199439
arXiv: https://arxiv.org/abs/2104.07431
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. 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.