# On contraction of vertices of the circuits in coset diagrams for \({\varvec{PSL}}\varvec{(2}, \pmb {\mathbb {Z}} \varvec{)}\)

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## Abstract

Coset diagrams for the action of \( PSL (2,\mathbb {Z})\) on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field \( F_{q}\), denoted by \( PL ( F_{q}) \), vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for \( PL ( F_{q}) \) is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by \(( xy) ^{m_{1}}( xy^{-1}) ^{m_{2}}\), that is, \(( m_{1},m_{2}) \). Let \(\alpha \) be the homomorphic image of \(( m_{1},m_{2}) \) obtained by contracting a pair of vertices *v*, *u* of \( ( m_{1},m_{2}) \). If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from \( \alpha \). In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of \(( m_{1},m_{2}) ?\) We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.

## Keywords

Modular group coset diagrams homomorphic images projective line over finite field## 2010 Mathematics Subject Classification

Primary: 05C25 Secondary: 20G40## References

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