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June 1999 A Characterization of Invertible Trace Maps Associated with a Substitution
Zhi-Xiong WEN, Zhi-Ying WEN
Tokyo J. Math. 22(1): 65-74 (June 1999). DOI: 10.3836/tjm/1270041612

Abstract

Let $F=\langle a,b\rangle$ be the free group generated by $a,b$. Let $\phi\in\mathrm{Hom}(F,SL(2,\mathbf{C}))$ be a homomorphism from $F$ to $SL(2,\mathbf{C})$. Define $T(\phi)=(\mathrm{tr}\,\phi(a),\mathrm{tr}\,\phi(b),\mathrm{tr}\,\phi(ab))$, where $\mathrm{tr}\,A$ stands for the trace of the matrix $A$. Let $\sigma\in\mathrm{Aut}F$. Then from [2, 12, 4], there exists a unique polynomial map $\Phi_{\sigma}\in(\mathbf{Z}[x,y,x])^3$, such that \[ \mathrm{tr}\,\phi(\sigma(a)),\mathrm{tr}\,\phi(\sigma(b)),\mathrm{tr}\,\phi(\sigma(ab)))=\Phi_{\sigma}(\mathrm{tr}\,\phi(a),\mathrm{tr}\,\phi(b),\mathrm{tr}\,\phi(ab)) \] with $x=\mathrm{tr}\,\phi(a),y=\mathrm{tr}\,\phi(b),z=\mathrm{tr}\,\phi(ab)$, and there exists a unique polynomial $Q_{\sigma}$, such that $\lambda\circ\Phi_{\sigma}=\lambda\cdot Q_{\sigma}$, where $\lambda(x,y,z)=x^2+y^2+z^2-xyz-4$. In this paper, we will show that $\sigma\in\mathrm{Aut}F$ if and only if $Q_{\sigma}(2,2,z)\equiv 1$, and that this result cannot be improved.

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Zhi-Xiong WEN. Zhi-Ying WEN. "A Characterization of Invertible Trace Maps Associated with a Substitution." Tokyo J. Math. 22 (1) 65 - 74, June 1999. https://doi.org/10.3836/tjm/1270041612

Information

Published: June 1999
First available in Project Euclid: 31 March 2010

zbMATH: 0987.11012
MathSciNet: MR1692020
Digital Object Identifier: 10.3836/tjm/1270041612

Rights: Copyright © 1999 Publication Committee for the Tokyo Journal of Mathematics

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Vol.22 • No. 1 • June 1999
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