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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Induced automorphisms on Fricke characters of free groups


Author: Robert D. Horowitz
Journal: Trans. Amer. Math. Soc. 208 (1975), 41-50
MSC: Primary 20E35; Secondary 10D10
DOI: https://doi.org/10.1090/S0002-9947-1975-0369540-8
MathSciNet review: 0369540
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0369540-8
Keywords: Automorphisms of a free group, character under a general or indeterminate representation in the group of $ 2 \times 2$ matrices with determinant 1, induced automorphisms on the ring of polynomial expressions with integer coefficients in the characters
Article copyright: © Copyright 1975 American Mathematical Society