On Certain Elements of Free Groups

S. V. Ivanov

doi:10.1006/jabr.1997.7354

On Certain Elements of Free Groups

S. V. Ivanov

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

LetF_mbe a free group of finite rankm. It is proven that for everyn≥2 there is a non-trivial wordw_n(x₁,...,x_n) such that if valuesw_n(U_n),w_n(V_n) ofw_n(x₁,...,x_n) on twon-tuplesU_nandV_nof elements ofF_mare conjugate and non-trivial then thesen-tuples themselves are conjugate. As a corollary, one has the existence of two elements inF_mwhose images uniquely determine any monomorphism ψ:F_m→F_m.

Original language	English (US)
Pages (from-to)	394-405
Number of pages	12
Journal	Journal of Algebra
Volume	204
Issue number	2
DOIs	https://doi.org/10.1006/jabr.1997.7354
State	Published - Jun 15 1998

ASJC Scopus subject areas

Algebra and Number Theory

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10.1006/jabr.1997.7354

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title = "On Certain Elements of Free Groups",

abstract = "LetFmbe a free group of finite rankm. It is proven that for everyn≥2 there is a non-trivial wordwn(x1,...,xn) such that if valueswn(Un),wn(Vn) ofwn(x1,...,xn) on twon-tuplesUnandVnof elements ofFmare conjugate and non-trivial then thesen-tuples themselves are conjugate. As a corollary, one has the existence of two elements inFmwhose images uniquely determine any monomorphism ψ:Fm→Fm.",

author = "Ivanov, {S. V.}",

note = "Funding Information: * Supported in part by an Alfred P. Sloan Research Fellowship, a Beckman Fellowship, and NSF Grant DMS 95-01056.",

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PY - 1998/6/15

Y1 - 1998/6/15

N2 - LetFmbe a free group of finite rankm. It is proven that for everyn≥2 there is a non-trivial wordwn(x1,...,xn) such that if valueswn(Un),wn(Vn) ofwn(x1,...,xn) on twon-tuplesUnandVnof elements ofFmare conjugate and non-trivial then thesen-tuples themselves are conjugate. As a corollary, one has the existence of two elements inFmwhose images uniquely determine any monomorphism ψ:Fm→Fm.

AB - LetFmbe a free group of finite rankm. It is proven that for everyn≥2 there is a non-trivial wordwn(x1,...,xn) such that if valueswn(Un),wn(Vn) ofwn(x1,...,xn) on twon-tuplesUnandVnof elements ofFmare conjugate and non-trivial then thesen-tuples themselves are conjugate. As a corollary, one has the existence of two elements inFmwhose images uniquely determine any monomorphism ψ:Fm→Fm.

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On Certain Elements of Free Groups

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