Discrete groups of slow subgroup growth |
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Authors: | Alexander Lubotzky László Pyber Aner Shalev |
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Institution: | 1. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel 2. Mathematical Institute, Hungarian Academy of Science, P.O.B. 127, H-1364, Budapest, Hungary 3. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
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Abstract: | It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least $n^{\frac{{\log n}}{{\log \log n}}} $ . In this paper we prove the existence of a finitely generated group whose subgroup growth is of type $n^{\frac{{\log n}}{{(\log \log n)^2 }}} $ . This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen logn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostn c logn subgroups of indexn. |
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