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

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.