Homomorphic Images of Circuits in $$\hbox {PSL}(2,{\mathbb {Z}})$$ PSL ( 2 , Z ) -Space

Abstract

Each conjugacy class of actions of \(PGL\left( {2,{\mathbb {Z}}} \right) \) on the projective line over a finite field \(F_q \) denoted by \(PL\left( {F_q } \right) \), can be represented by a coset diagram \(D\left( {\theta ,q} \right) \), where \(\theta \in F_q \) and q is a prime power. The coset diagrams are composed of fragments, and the fragments are further composed of two or more circuits at a certain common point. Professor Graham Higman raised a question: for what values of q and \(\theta \), can a fragment \(\gamma \) be found in \(D\left( {\theta ,q} \right) ?\) Mushtaq in 1983 found that the condition for the existence of a fragment in \(D\left( {\theta ,q} \right) \) is a polynomial f in \({\mathbb {Z}}\left[ z \right] \). In this paper, we answer the question: how many polynomials are obtained from the fragments, composed by joining the circuits \(\left( {n_1 ,n_2 } \right) \) and \(\left( {m_1 ,m_2 } \right) \), where \(n_2 <n_1 <m_2 <m_1\), at all points of connection.