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Economical separability in free groups

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Abstract

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 2.1.1.419).

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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). https://doi.org/10.1007/s11202-009-0067-7

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Keywords

  • separability by a subgroup