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Licensed Unlicensed Requires Authentication Published by De Gruyter October 17, 2017

Normal subgroups in limit groups of prime index

  • Thomas S. Weigel EMAIL logo and Jhoel S. Gutierrez
From the journal Journal of Group Theory

Abstract

Motivated by their study of pro-p limit groups, D. H. Kochloukova and P. A. Zalesskii formulated in [15, Remark after Theorem 3.3] a question concerning the minimum number of generators d(N) of a normal subgroup N of prime index p in a non-abelian limit group G (see Question*). It is shown that the analogous question for the rational rank has an affirmative answer (see Theorem A). From this result one may conclude that the original question of Kochloukova and Zalesskii has an affirmative answer if the abelianization Gab of G is torsion free and d(G)=d(Gab) (see Corollary B), or if G is a special kind of one-relator group (see Theorem D).


Communicated by Dessislava H. Kochloukova


Funding statement: The second author was supported by CNPq-Brazil.

Acknowledgements

The authors would like to thank H. Wilton for a very useful comment concerning an earlier version of the paper, and also the referee for his/her helpful remarks.

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Received: 2016-7-25
Revised: 2017-5-25
Published Online: 2017-10-17
Published in Print: 2018-1-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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