Groups acting on rooted trees

In the developing structure theory of totally disconnected locally compact groups, compact open subgroups play an important role. Specifically, certain compact open subgroups called tidy subgroups that act in an analogous way to eigenspaces in linear algebra. Supposing V is a tidy subgroup for the automorphism α, there is a corresponding closed subgroup V₋₋ that is invariant under α. This gives rise to a representation of V₋₋ ⋊ 〈α〉 as a closed subgroup of the stabiliser of an end of an infinite homogeneous tree. The problem of understanding this representation can be reduced to studying groups satisfying an additional condition called property R that act on a regular rooted tree. As these groups are profinite we instead look at groups with a finite version of property R_n.

In this paper we explicitly calculate some examples for the ternary tree while simultaneously establishing results about abelian groups satisfying property R_n. We show that every abelian group satisfying property R_n describes a regular ac­tion on the n-th level of the corresponding rooted tree. Using this we develop a construction to classify all such abelian groups acting on a p-ary tree of any depth where p is prime. A construction that uses existing groups to provide further examples of groups satisfying property R_n on higher degree trees is also outlined.