Left–right pairs and complex forests of infinite rooted binary trees

Abstract

Let [Formula: see text], and let [Formula: see text] be a pair of Möbius transformations corresponding to [Formula: see text] matrices such that [Formula: see text] and [Formula: see text] are disjoint. Given such a pair (called a left–right pair), we can construct a directed graph [Formula: see text] with vertices [Formula: see text] and edges [Formula: see text], which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of [Formula: see text] whose corresponding Möbius transformations form left–right pairs and showing that trees in [Formula: see text] are always rooted.