Abstract
Granted the three integers n ≥ 2, r, and R, consider all ordered tuples of r elements of length at most R in the free group F n . Calculate the number of those tuples that generate in F n a rank r subgroup and divide it by the number of all tuples under study. As R → ∞, the limit of the ratio is known to exist and equal 1 (see [1]). We give a simple proof of this result.
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References
Martino A., Turner E. C., and Ventura E., “The density of injective endomorphisms of a free group,” [Preprint, No. 685 of CRM] (2006). Available at: http://www.crm.cat.
Lyndon R. C. and Schupp P. E., Combinatorial Group Theory. Vol. 2 [Russian translation], Mir, Moscow (1980).
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Original Russian Text Copyright © 2009 Buskin N. V.
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Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 289–291, March–April, 2009.
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Buskin, N.V. The probability that r elements of a rank n free group generate a rank r subgroup. Sib Math J 50, 231–232 (2009). https://doi.org/10.1007/s11202-009-0026-3
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DOI: https://doi.org/10.1007/s11202-009-0026-3