G. N. Arzhantseva, A. Yu. Ol'shanskii, “The class of groups all of whose subgroups with lesser number of generators are free is generic”, Mat. Zametki, 59:4 (1996), 489–496; Math. Notes, 59:4 (1996), 350

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Matematicheskie Zametki, 1996, Volume 59, Issue 4, Pages 489–496
DOI: https://doi.org/10.4213/mzm1744 (Mi mzm1744)

This article is cited in 75 scientific papers (total in 75 papers)

The class of groups all of whose subgroups with lesser number of generators are free is generic

G. N. Arzhantseva, A. Yu. Ol'shanskii

M. V. Lomonosov Moscow State University

Full-text PDF (351 kB) Citations (75)

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DOI: https://doi.org/10.4213/mzm1744

Abstract: It is shown that, in a certain statistical sense, in almost every group with

m

$m$ generators and

n

$n$ relations (with

m

$m$ and

n

$n$ chosen), any subgroup generated by less than

m

$m$ elements (which need not belong to the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked groups.

Received: 05.01.1995

English version:
Mathematical Notes, 1996, Volume 59, Issue 4, Pages 350–355
DOI: https://doi.org/10.1007/BF02308683

Bibliographic databases:

UDC: 512.5

Language: Russian

Citation: G. N. Arzhantseva, A. Yu. Ol'shanskii, “The class of groups all of whose subgroups with lesser number of generators are free is generic”, Mat. Zametki, 59:4 (1996), 489–496; Math. Notes, 59:4 (1996), 350–355

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm1744

https://doi.org/10.4213/mzm1744

https://www.mathnet.ru/eng/mzm/v59/i4/p489

This publication is cited in the following 75 articles:

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Kapovich I., “Primitivity Rank For Random Elements in Free Groups”, J. Group Theory, 2022
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Olga Kharlampovich, Pascal Weil, Lecture Notes in Computer Science, 12180, Fields of Logic and Computation III, 2020, 147
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Gupta N., Kapovich I., “The Primitivity Index Function For a Free Group, and Untangling Closed Curves on Hyperbolic Surfaces. With the Appendix By Khalid Bou-Rabee”, Math. Proc. Camb. Philos. Soc., 166:1 (2019), 83–121
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Gekhtman I., Taylor S.J., Tiozzo G., “Counting Loxodromics For Hyperbolic Actions”, J. Topol., 11:2 (2018), 379–419
Cavaleri M., “Folner Functions and the Generic Word Problem For Finitely Generated Amenable Groups”, J. Algebra, 511 (2018), 388–404
Brown R.F., Nan J., “Stabilizers of Fixed Point Classes and Nielsen Numbers of N-Valued Maps”, Bull. Belg. Math. Soc.-Simon Steven, 24:4, SI (2017), 523–535

Citing articles in Google Scholar: Russian citations, English citations
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