Non-freeness of parabolic two-generator groups

Abstract

A complex number $\lambda$ is said to be non-free if the subgroup of $SL(2,\bc)$ generated by $$X=\begin{pmatrix} 1& 1\\ 0 & 1 \end{pmatrix} \,\, \text{and}\,\,\,Y_{\lambda}=\begin{pmatrix} 1& 0\\ \lambda & 1 \end{pmatrix}$$ is not a free group of rank 2. In this case the number $\lambda$ 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 $\lambda$ being a relation number and $u:=\sqrt{- \lambda}$ being a root of a `generalized Chebyshev polynomial'. The generalized Chebyshev polynomials of degree $k$ are given by a sequence of $k$ integers $(n_1, n_2,\cdots, n_k)$ 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 $\lambda$ is a relation number of $u$-degree $k$ 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 $k$. 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 $(-4, 4)$ is a relation number.