Induced automorphisms on Fricke characters of free groups

Horowitz, Robert D.

doi:10.1090/S0002-9947-1975-0369540-8

Induced automorphisms on Fricke characters of free groups
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Trans. Amer. Math. Soc. 208 (1975), 41-50 Request permission

Abstract:

The term character in this paper will denote the character of a group element under a general or indeterminate representation of the group in the special linear group of $2 \times 2$ matrices with determinant 1; the properties of characters of this type were first studied by R. Fricke in the late nineteenth century. Theorem 1 determines the automorphisms of a free group which leave the characters invariant. In a previous paper it was shown that the character of each element in the free group ${F_n}$ of finite rank $n$ can be identified with an element of a certain quotient ring of the commutative ring of polynomials with integer coefficients in ${2^n} - 1$ indeterminates. It follows that any automorphism of ${F_n}$ induces in a natural way an automorphism on this quotient ring. Corollary 1 shows that for $n \geqslant 3$ the group of induced automorphisms of ${F_n}$ is isomorphic to the group of outer automorphism classes of ${F_n}$. The possibility is thus raised that the induced automorphisms may be useful in studying the structure of this group. Theorem 2 gives a characterization for the group of induced automorphisms of ${F_2}$ in terms of an invariant polynomial.

References

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