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Fundam. Prikl. Mat., 1997, Volume 3, Issue 3, Pages 675–683 (Mi fpm237)  

This article is cited in 15 scientific papers (total in 15 papers)

On the groups in which the subgroups with fixed number of generators are free

G. N. Arzhantseva

M. V. Lomonosov Moscow State University

Abstract: We prove here that, in a definite statistical meaning, in almost every group with $m$ generators and $n$ relations (we suppose $m$ and $n$ to be fixed) all $\le L$-generated subgroups of infinite index are free ($L$ is an arbitrary preassigned bound, possibly $L\gg m$) and all subgroups of finite index are not free. To prove this fact we found the condition on relations which guarantee that all subgroups of infinite index with fixed number of generators in a finitely presented group are free. This condition is formulated by means of the finite marked graphs.

Full text: PDF file (417 kB)

Bibliographic databases:
UDC: 519.40
Received: 01.01.1996

Citation: G. N. Arzhantseva, “On the groups in which the subgroups with fixed number of generators are free”, Fundam. Prikl. Mat., 3:3 (1997), 675–683

Citation in format AMSBIB
\Bibitem{Arz97}
\by G.~N.~Arzhantseva
\paper On the groups in which the subgroups with fixed number of generators are free
\jour Fundam. Prikl. Mat.
\yr 1997
\vol 3
\issue 3
\pages 675--683
\mathnet{http://mi.mathnet.ru/fpm237}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1794135}
\zmath{https://zbmath.org/?q=an:0929.20025}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Arzhantseva, GN, “Generic properties of finitely presented groups and Howson's theorem”, Communications in Algebra, 26:11 (1998), 3783  crossref  mathscinet  zmath  isi
    2. Arzhantseva, GN, “A property of subgroups of infinite index in a free group”, Proceedings of the American Mathematical Society, 128:11 (2000), 3205  crossref  mathscinet  zmath  isi
    3. Bumagin, I, “On small cancellation k-generated groups with (k-1)-generated subgroups all free”, International Journal of Algebra and Computation, 11:5 (2001), 507  crossref  mathscinet  zmath  isi
    4. Kapovich, I, “Generic-case complexity, decision problems in group theory, and random walks”, Journal of Algebra, 264:2 (2003), 665  crossref  mathscinet  zmath  isi
    5. Ghys, T, “Random groups (after Misha Gromov,...)”, Asterisque, 2004, no. 294, 173  mathscinet  zmath  isi
    6. Kapovich, I, “Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion”, Proceedings of the London Mathematical Society, 88 (2004), 89  crossref  mathscinet  zmath  isi
    7. Kapovich, I, “Delzant's T-invariant, Kolmogorov complexity and one-relator groups”, Commentarii Mathematici Helvetici, 80:4 (2005), 911  crossref  mathscinet  zmath  isi  elib
    8. Kapovich, I, “Genericity, the Arzhantseva-Ol'shanskii method and the isomorphism problem for one-relator groups”, Mathematische Annalen, 331:1 (2005), 1  crossref  mathscinet  zmath  isi  elib
    9. Kapovich, I, “Average-case complexity and decision problems in group theory”, Advances in Mathematics, 190:2 (2005), 343  crossref  mathscinet  zmath  isi
    10. Kapovich, I, “Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups”, Pacific Journal of Mathematics, 223:1 (2006), 113  crossref  mathscinet  zmath  isi
    11. Kapovich, I, “Densities in free groups and Z(k) visible points and test elements”, Mathematical Research Letters, 14:2–3 (2007), 263  crossref  mathscinet  zmath  isi  elib
    12. Kaimanovich, V, “The subadditive ergodic theorem and generic stretching factors for free group automorphisms”, Israel Journal of Mathematics, 157:1 (2007), 1  crossref  mathscinet  zmath  isi
    13. Kapovich, I, “On group-theoretic models of randomness and genericity”, Groups Geometry and Dynamics, 2:3 (2008), 383  crossref  mathscinet  zmath  isi
    14. Kapovich, I, “Random quotients of the modular group are rigid and essentially incompressible”, Journal fur Die Reine und Angewandte Mathematik, 628 (2009), 91  crossref  mathscinet  zmath  isi
    15. V. N. Bezverkhnii, I. V. Dobrynina, “O svobodnykh podgruppakh v gruppakh Artina s drevesnoi strukturoi”, Chebyshevskii sb., 15:1 (2014), 32–42  mathnet
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