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 and a prime , the upper -rank of is the supremum of the sectional -ranks of all finite quotients of . It is unknown whether, for a finitely generated group , these numbers can be finite but unbounded as ranges over all primes. The conjecture that this cannot happen if is soluble is reduced to an analogous `relative' conjecture about the upper -ranks of a `quasi-finitely-generated' module for a soluble minimax group . The main result establishes a special case of this relative conjecture, namely when the module is finitely generated and the minimax group 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