Forests of complex numbers

Abstract

The Calkin–Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form [Formula: see text], where [Formula: see text] and [Formula: see text] are relatively prime positive integers. In this paper, certain subsemigroups of the modular group are used to construct similar trees in the set [Formula: see text] of positive complex numbers. Associated to each subsemigroup is a forest of trees that partitions [Formula: see text]. The fundamental domain and the set of cusps of the subsemigroup are defined and computed.