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Triangles of Baumslag–Solitar Groups

Published online by Cambridge University Press:  20 November 2018

Daniel Allcock
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Abstract

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Our main result is that many triangles of Baumslag–Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag–Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length $\le \,3$. We obtain a lot of information about finite quotients, even when we cannot determine developability.

Keywords

20F0620F65
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 241 - 253
Copyright
Copyright © Canadian Mathematical Society 2012

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