A sequence of algebraic integer relation numbers which converges to 4

2020/10/19 by Wonyong Jang, Jang, Wonyong, KyeongRo Kim +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2010.09560

openalex publication_date 2020/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let α∈ ℝ and let A=\beginbmatrix 1 amp; 1
0 amp; 1\endbmatrix and B_α = \beginbmatrix 1 amp; 0
αamp; 1\endbmatrix. The subgroup G_α of SL₂(ℝ) is a group generated by the matrices A and B_α. In this paper, we investigate the property of the group G_α. We construct a generalization of the Farey graph for the subgroup G_α. This graph determines whether the group G_α is a free group of rank 2. More precisely, the group G_α is a free group of rank 2 if and only if the graph is tree. In particular, we show that if 1/2 is a vertex of the graph, then G_α is not a free group of rank 2. Using this, we construct a sequence of real numbers so that the sequence converges to 4 and each number has the corresponding group that is not a free group of rank 2. It turns out that the real numbers are algebraic integers.

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• On non-freeness of groups generated by two parabolic matrices with rational parameters: limit points and the orbit test

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