Non-freeness of parabolic two-generator groups
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| Format: | Preprint |
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2024
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| _version_ | 1866916289761509376 |
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| author | Choi, Philip Jo, Kyeonghee Kim, Hyuk Lee, Junho |
| author_facet | Choi, Philip Jo, Kyeonghee Kim, Hyuk Lee, Junho |
| contents | A complex number $λ$ is said to be non-free if the subgroup of $SL(2,\bc)$ generated by $$X=\begin{pmatrix} 1& 1\\ 0 & 1
\end{pmatrix} \,\, \text{and}\,\,\,Y_λ=\begin{pmatrix} 1& 0\\ λ& 1
\end{pmatrix}$$ 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 $u:=\sqrt{- λ}$ being a root of a `generalized Chebyshev polynomial'. The generalized Chebyshev polynomials of degree $k$ are given by a sequence of $k$ integers $(n_1, n_2,\cdots, n_k)$ 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 $λ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-freeness of parabolic two-generator groups Choi, Philip Jo, Kyeonghee Kim, Hyuk Lee, Junho Group Theory Geometric Topology 20E05, 11B39, 11J70, 30F35, 30F40 A complex number $λ$ is said to be non-free if the subgroup of $SL(2,\bc)$ generated by $$X=\begin{pmatrix} 1& 1\\ 0 & 1 \end{pmatrix} \,\, \text{and}\,\,\,Y_λ=\begin{pmatrix} 1& 0\\ λ& 1 \end{pmatrix}$$ 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 $u:=\sqrt{- λ}$ being a root of a `generalized Chebyshev polynomial'. The generalized Chebyshev polynomials of degree $k$ are given by a sequence of $k$ integers $(n_1, n_2,\cdots, n_k)$ 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 $λ$ 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. |
| title | Non-freeness of parabolic two-generator groups |
| topic | Group Theory Geometric Topology 20E05, 11B39, 11J70, 30F35, 30F40 |
| url | https://arxiv.org/abs/2406.11378 |