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You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.

Edit: A quick search found the paper http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf which confirms Andreas comment. It also gives the action of $Out(F_2)$ in Example 6.1. This raises the question: does $Out(F_2)$ act transitively on this set of polynomials?

In particular they give the reference:

Fricke, R., and Klein, F., Vorlesungen uber die Theorie der automorphem Functionen, Vol. 1,
pp. 365-370. Leipzig: B.G. Teubner 1897. Reprint: New York Juhnson Reprint Corporation
(Academic Press) 1965.

You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.

Edit: A quick search found the paper http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf which confirms Andreas comment. It also gives the action of $Out(F_2)$ in Example 6.1. This raises the question: does $Out(F_2)$ act transitively on this set of polynomials?

In particular they give the reference:

`Fricke, R., and Klein, F., Vorlesungen uber die Theorie der automorphem Functionen, Vol. 1, pp. 365-370. Leipzig: B.G. Teubner 1897. Reprint: New York Juhnson Reprint Corporation (Academic Press) 1965

This can be found at http://archive.org/details/vorlesungenber01fricuoft

You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.

Edit: A quick search found the paper http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf which confirms Andreas comment. It also gives the action of $Out(F_2)$ in Example 6.1. This raises the question: does $Out(F_2)$ act transitively on this set of polynomials?

You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.

Edit: A quick search found the paper http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf which confirms Andreas comment. It also gives the action of $Out(F_2)$ in Example 6.1. This raises the question: does $Out(F_2)$ act transitively on this set of polynomials?

In particular they give the reference:

Fricke, R., and Klein, F., Vorlesungen uber die Theorie der automorphem Functionen, Vol. 1,
pp. 365-370. Leipzig: B.G. Teubner 1897. Reprint: New York Juhnson Reprint Corporation
(Academic Press) 1965.

You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.

You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.

Edit: A quick search found the paper http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf which confirms Andreas comment. It also gives the action of $Out(F_2)$ in Example 6.1. This raises the question: does $Out(F_2)$ act transitively on this set of polynomials?