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A remark on thickness of free-by-cyclic groups. (English) Zbl 1515.20222

Summary: Let \(F\) be a free group of positive, finite rank and let \(\Phi \in \operatorname{Aut}(F)\) be a polynomial-growth automorphism. Then \(F\rtimes_\Phi\mathbb{Z}\) is strongly thick of order \(\eta\), where \(\eta\) is the rate of polynomial growth of \(\phi\). This fact is implicit in work of N. Macura [Q. J. Math. 53, No. 2, 207–239 (2002; Zbl 1036.20033)], whose results predate the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth of and thickness explicit. Our result combines with a result independently due to F. Dahmani and R. Li [“Relative hyperbolicity for of free products”, Preprint, arXiv:1901.06760], F. Gautero and M. Lustig [“The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth”, Preprint, arXiv:0707.0822], and P. Ghosh [Compos. Math. 159, No. 1, 153–183 (2023; Zbl 1522.20173)] to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgroups.

MSC:

20F65 Geometric group theory
20E05 Free nonabelian groups
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
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References:

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