Abstract
The first version (historically) of the PSG Theorem to be proved dealt with the very special case of groups that are assumed to be both soluble and residually nilpotent; under these hypotheses, an elementary argument sufficed for the proof, involving none of the sophisticated mathematics that we have seen in Chapter 5. In Section 2 of this chapter we give a slightly more sophisticated (though still elementary) proof of the following sharper result: Theorem 9.1 Let G be a finitely generated virtually soluble group that is virtually residually nilpotent. Then either G has finite rank (and hence PSG), or there exist c > 1 and \(d \in \mathbb{N}\) such that \({S_n}(G) \geqslant S_n^{ \triangleleft \triangleleft }(G) \geqslant {c^{{n^{1/d}}}}\) for all large n.
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© 2003 Birkhäuser Verlag
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Lubotzky, A., Segal, D. (2003). Soluble Groups. In: Subgroup Growth. Progress in Mathematics, vol 212. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8965-0_10
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DOI: https://doi.org/10.1007/978-3-0348-8965-0_10
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9846-1
Online ISBN: 978-3-0348-8965-0
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