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