^{1}

Higman has defined coset diagrams for

The modular group

The Hecke group

If

There is a well-known relationship between the action of

In

Two homomorphisms

By a circuit in a coset diagram for an action of

Consider two non-periodic and simple circuits

We connect the vertex

One can see that the vertex

The action of

The coset diagram

Given a fragment, there is a polynomial

if the fragment occurs in

if

In [

This gives a homogeneous equation in

Two pairs of words

In this paper, we find distinct pairs of words that are equivalent. We also show there are certain fragments, which have the same orientations as those of their mirror images.

In the above notation, the polynomial obtained from a fragment

Let

If a vertex

We know that

Let fragment

Suppose vertex

Since

Let

As in Example

The number of points of connection of the circuits for a fragment is always multiple of three and plays a significant role because they are directly related to the structure of the fragment. The following theorem illustrates this relation.

Let

Since circuits

Consider a fragment

The following Theorem shows that

The following pairs of words are equivalent:

Let

Similarly, it is easy to prove that

Theorem

In a coset diagram for the action of

Let

Since

Now, by using (

If a vertex

Consider a simple circuit

In Figure

The vertices

Let the homomorphic image of the fragment (Figure

If the fragment

Let

The polynomials obtained from the fragment

Let

Since a unique polynomial is obtained from a fragment

Consider two circuits

(i) The fragment composed by joining a vertex

(ii) The fragment composed by joining a vertex

(iii) The fragment composed by joining a vertex

(i) Let

(ii) Let

(iii) Let

If we join a pair of circuits at a certain point, we get a fragment and hence a polynomial. Since each such polynomial splits linearly in a suitable Galois [

How many fragments (polynomials) are obtained if we join a pair of circuits

In order to give the answer of the above question, we first have to find those pair of words which are equivalent. In other words, we have to find those points of connection, at which we get the same polynomial. This issue is resolved in this paper. We will give the answer of the above question in an other paper. After that, one can establish a connection between a class of groups and a pair of circuits

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the referee for his very helpful comments, which have significantly improved the presentation of this paper.