Vestnik TVGU. Seriya: Prikladnaya Matematika [Herald of Tver State University. Series: Applied Mathematics]
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Vestnik TVGU. Seriya: Prikladnaya Matematika [Herald of Tver State University. Series: Applied Mathematics], 2022, Issue 2, Pages 5–13
DOI: https://doi.org/10.26456/vtpmk633 (Mi vtpmk633) 		 
Theoretical Computer Science, Cybernetics

Violation of finite approximability for single-coefficient commutant equations

V. G. Durneva, A. I. Zetkinaab

a P.G. Demidov Yaroslavl State University, Yaroslavl
b Voronezh State University, Voronezh
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DOI: https://doi.org/10.26456/vtpmk633
Abstract: We construct an equation (resolved with respect to unknowns) such that (i) it has no solution in $F_2$ (a free group of rank $2$), but (ii) it has a solution in any finite homomorphic image of $F_2$. The left-hand-side of this equation belongs to the derived subgroup (i.e. has zero sum of exponents in each variable), while its right-hand-side is the commutator of two generators of $F_2$.
Keywords: free group, equation in a free group, residual finiteness, commutator of elements, commutator subgroup.
Funding agency Grant number
Russian Foundation for Basic Research 19-52-26006-а
Received: 21.04.2022
Revised: 18.05.2022
Bibliographic databases:
Document Type: Article
UDC: 512.54.05, 512.543.7
Language: Russian
Citation: V. G. Durnev, A. I. Zetkina, “Violation of finite approximability for single-coefficient commutant equations”, Vestnik TVGU. Ser. Prikl. Matem. [Herald of Tver State University. Ser. Appl. Math.], 2022, no. 2, 5–13
Citation in format AMSBIB
\Bibitem{DurZet22}
\by V.~G.~Durnev, A.~I.~Zetkina
\paper Violation of finite approximability for single-coefficient commutant equations
\jour Vestnik TVGU. Ser. Prikl. Matem. [Herald of Tver State University. Ser. Appl. Math.]
\yr 2022
\issue 2
\pages 5--13
\mathnet{http://mi.mathnet.ru/vtpmk633}
\crossref{https://doi.org/10.26456/vtpmk633}
\elib{https://elibrary.ru/item.asp?id=49249005}
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