, Volume 96, Issue 2, pp 399-418

Discrete groups of slow subgroup growth

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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.

The first and third authors acknowledge support of The Israel Science Foundation, administered by The Israel Academy of Sciences and Humanities. The second author acknowledges support of the Hungarian National Foudation for Scientific Research, Grant No. T7441.