Left-Right Pairs and Complex Forests of Infinite Rooted Binary Trees

Abstract

Let ${\mathcal{D}}_0:=\{x+iy|x,y>0\}$, and let $(L,R)$ be a pair of Möbius transformations corresponding to ${\mathrm{SL}}_2({\mathbb{N}}_0)$ matrices such that $R({\mathcal{D}}_0)$ and $L({\mathcal{D}}_0)$ are disjoint. Given such a pair (called a left-right pair), we can construct a directed graph $\mathcal{F}(L,R)$ with vertices ${\mathcal{D}}_0$ and edges $\{(z,R(z)){\}}_{z\in {\mathcal{D}}_0}\cup \{(z,L(z)){\}}_{z\in {\mathcal{D}}_0}$, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of ${\mathrm{SL}}_2({\mathbb{N}}_0)$ whose corresponding Möbius transformations form left-right pairs and showing that trees in $\mathcal{F}(L,R)$ are always rooted.

8 pages