1. Non-freeness of parabolic two-generator groups.
- Author
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Choi, Philip, Jo, Kyeonghee, Kim, Hyuk, and Lee, Junho
- Subjects
- *
CHEBYSHEV polynomials , *CONTINUED fractions , *COMPLEX numbers , *FREE groups , *CHECKS - Abstract
A complex number λ is said to be non-free if the subgroup of SL (2 , ℂ) generated by X = 1 1 0 1 and Y λ = 1 0 λ 1 is not a free group of rank 2. In this case, the number λ 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 λ being a relation number and u : = − λ 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 , ... , 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 λ 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. [ABSTRACT FROM AUTHOR]
- Published
- 2025
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