Hyperfiniteness for group actions on trees (Q6583742)

An open problem in geometric group theory is if all actions of amenable groups give rise to hyperfinite equivalence relations (see [\textit{T. Clinton} et al., Duke Math. J. 172, No. 16, 3175--3226 (2023; Zbl 07783739)]). The problem of identifying hyperfiniteness for natural actions associated to non-amenable groups is also of interest and is seen to be tractable in certain cases.\N\NThe main results in the paper under review are as follows.\N\NTheorem A: Let \(G\curvearrowright T\) be an action of a countable group on a countable tree, and suppose that every geodesic ray \(\mathbf{v}\) in \(T\) has an initial segment \(\sigma\) with \(\mathrm{Stab}(\sigma)=\mathrm{Stab}(\mathbf{v})\). Then the induced action of \(G\) on the Gromov boundary \(\partial T\) is Borel hyperfinite.\N\NTheorem B: Let \(G\curvearrowright T\) be an action of a countable group on a countable tree such that every geodesic ray \(\mathbf{v}\) has an initial segment \(\sigma\) such that \(\mathrm{Stab}(\mathbf{v})\) is uniformly coamenable (see Definition 4.3) in \(\mathrm{Stab}(\sigma)\). Then the induced action of \(G\) on the Gromov boundary \(\partial T\) is Borel \(2\)-amenable, so in particular measure-hyperfinite.\N\NA natural question arising from the above results is whether exist countable group actions on trees for which the induced equivalence relation on the Gromov boundary is not Borel hyperfinite. The authors, in a final section, answer this question affirmatively and identify natural examples arising from actions on trees of certain inverse limits of countable groups.