Non-freeness of parabolic two-generator groups

A complex number [Formula: see text] is said to be non-free if the subgroup of [Formula: see text] generated by [Formula: see text] is not a free group of rank 2. In this case, the number [Formula: see text] is called a relation number, and it has been a long standing problem to determine the relation numbers. In this paper, we characterize the relation numbers by establishing the equivalence between [Formula: see text] being a relation number and [Formula: see text] being a root of a ‘generalized Chebyshev polynomial’. The generalized Chebyshev polynomials of degree [Formula: see text] are given by a sequence of [Formula: see text] integers [Formula: see text] using the usual recursive formula, and thereby can be studied systematically using continuants and continued fractions. Such formulation, then, enables us to prove that, the question whether a given number [Formula: see text] is a relation number of [Formula: see text]-degree [Formula: see text] can be answered by checking only finitely many generalized Chebyshev polynomials. Based on these theorems, we design an algorithm deciding any given number is a relation number with minimal degree [Formula: see text]. With its computer implementation we provide a few sample examples, with a particular emphasis on the well-known conjecture that every rational number in the interval [Formula: see text] is a relation number.