Growth in higher Baumslag–Solitar groups
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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.
KeywordsGrowth Baumslag–Solitar groups Solvable groups Rationality
Mathematics Subject Classification20F65 20F10 20F16 68Q45
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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