Consider the rank n free group F n with basis X. Bogopol’skiĭ conjectured in [1, Problem 15.35] that each element w ∈ F n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F 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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Kourovka Notebook: Unsolved Problems of Group Theory [in Russian], 15th edit., Inst. Mat. (Novosibirsk), Novosibirsk (2002).
Khalid Bou-Rabee, “Quantifying residual finiteness,” Technical Report arXiv:0807.0862v2 [math.GR], arxiv.org, 2008.
Rivin I., “Geodesics with one self-intersection, and other stories,” Technical Report arXiv:0901.2543v3 [math.GT], arxiv.org, 2009.
Malestein J. and Putman A., “On the self-intersections of curves deep in the lower central series of a surface group,” arXiv, math.GT, January 2009; 12 pages and 2 figures.
Bogopol’skiĭ O. V., Introduction to the Theory of Groups [in Russian], Institute of Computer Science, Moscow; Izhevsk (2002).
Ireland K. and Rosen M., A Classical Introduction to Modern Number Theory, Springer-Verlag, New York etc. (1990).
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 126.96.36.1999).
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 4, pp. 765–771, July–August, 2009.
About this article
Cite this article
Buskin, N.V. Economical separability in free groups. Sib Math J 50, 603–608 (2009). https://doi.org/10.1007/s11202-009-0067-7
- separability by a subgroup