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Geometriae Dedicata

, Volume 195, Issue 1, pp 79–99 | Cite as

Growth in higher Baumslag–Solitar groups

  • Ayla P. Sánchez
  • Michael Shapiro
Original Paper
  • 108 Downloads

Abstract

We study the HNN extension of \(\mathbb Z^m\) given by the cubing endomorphism \(g\mapsto g^3\), and prove that such groups have rational growth with respect to the standard generating sets. We compute the subgroup growth series of the horocyclic subgroup \(\mathbb Z^m\) in this family of examples, prove that for each m the subgroup has rational growth. We then use the tree-like structure of these groups to see how to compute the growth of the whole group.

Keywords

Growth Baumslag–Solitar groups Solvable groups Rationality 

Mathematics Subject Classification

20F65 20F10 20F16 68Q45 

Notes

Acknowledgements

The authors would like to thank Moon Duchin, Murray Elder, and Meng-Che Ho. MS would like to thank Karen Buck for helping to jump-start some of the neurons used in this work.

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Wheaton CollegeNortonUnited States
  2. 2.University of BathBathUK

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