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.
Video
The video of a lecture given by Robin Tucker-Drob about
this article:
http://library.cirm-math.fr/Record.htm?idlist=1&record=19391396124911195789
Recording on May 23/05/2023
during the thematic meeting : « Measured Group Theory,
Stochastic Processes on Groups and Borel Combinatorics » May 23,
2023 at the Centre International de Rencontres Mathématiques
(Marseille, France)
Abstract of the lecture:
We show that several new classes of groups are
measure strongly treeable, i.e., all of their free
measure-class-preserving actions are treeable. This includes
all finitely generated groups admitting planar Cayley graphs,
all finitely generated elementarily free groups, and more
generally all groups arising as the fundamental group of an
'IFL tower' over these groups. Our techniques also lead to new
measure strong free factors of groups, i.e., group elements
which generate a primitive subrelation in every free
measure-class-preserving action. This is based on joint work
with Clinton Conley, Damien Gaboriau, and Andrew Marks.
(Also on Youtube https://www.youtube.com/watch?v=lHSG2SlBgdI)