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On modules of finite upper rank


Author: Dan Segal
Journal: Trans. Amer. Math. Soc. 353 (2001), 391-410
MSC (2000): Primary 20C07, 20F16, 20E07
DOI: https://doi.org/10.1090/S0002-9947-00-02612-X
Published electronically: September 13, 2000
MathSciNet review: 1707703
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Abstract: For a group $G$ and a prime $p$, the upper $p$-rank of $G$ is the supremum of the sectional $p$-ranks of all finite quotients of $G$. It is unknown whether, for a finitely generated group $G$, these numbers can be finite but unbounded as $p$ ranges over all primes. The conjecture that this cannot happen if $G$ is soluble is reduced to an analogous `relative' conjecture about the upper $p$-ranks of a `quasi-finitely-generated' module $M$for a soluble minimax group $\Gamma$. The main result establishes a special case of this relative conjecture, namely when the module $M$ is finitely generated and the minimax group $\Gamma$ is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)


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Additional Information

Dan Segal
Affiliation: All Souls College, Oxford OX1 4AL, United Kingdom
Email: dan.segal@all-souls.ox.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-00-02612-X
Received by editor(s): March 3, 1999
Received by editor(s) in revised form: June 25, 1999
Published electronically: September 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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