Groups with fractionally exponential subgroup growth

Dan Segal*, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


We prove that, if the subgroup growth of a finitely generated metabelian group G is not polynomial, then it is at least cn 1 d for some positive integer d (where c>1 is a suitable constant). For each integer d>1 we construct a finitely presented metabelian group whose subgroup growth is approximately cn 1 d. Finally, we establish a sharp upper bound for the subnormal subgroup growth of finitely presented soluble groups, and derive a new necessary condition for a metabelian group to be finitely presented. Our methods involve results from algebraic geometry and the geometry of numbers, as well as Golod-Safarevic type inequalities.

Original languageAmerican English
Pages (from-to)205-223
Number of pages19
JournalJournal of Pure and Applied Algebra
Issue number1-3
StatePublished - 25 Aug 1993


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