Geodesic
Violation of finite approximability for single-coefficient commutant equations
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 2 (2022), pp. 5-13

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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. 
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     author = {V. G. Durnev and A. I. Zetkina},
     title = {Violation of finite approximability for single-coefficient commutant equations},
     journal = {Vestnik Tverskogo gosudarstvennogo universiteta. Seri\^a Prikladna\^a matematika},
     pages = {5--13},
     publisher = {mathdoc},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTPMK_2022_2_a0/}
}
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V. G. Durnev; A. I. Zetkina. Violation of finite approximability for single-coefficient commutant equations. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 2 (2022), pp. 5-13. http://geodesic.mathdoc.fr/item/VTPMK_2022_2_a0/