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Economical separability in free groups

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Consider the rank n free group F n with basis X. Bogopol’skiĭ conjectured in [1, Problem 15.35] that each element wF n of length |w| ≥ 2 with respect to X can be separated by a subgroup HF n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F n , as well as the separability by a subgroup of index at most |w|/2 + 2 in general.

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Author information

Correspondence to N. V. Buskin.

Additional information

Original Russian Text Copyright © 2009 Buskin N. V.

The author was supported by the Program “Development of the Scientific Potential of Higher School” of the Russian Federal Agency for Education (Grant


Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 4, pp. 765–771, July–August, 2009.

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Buskin, N.V. Economical separability in free groups. Sib Math J 50, 603–608 (2009).

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  • separability by a subgroup