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.


20F65 Geometric group theory
20E05 Free nonabelian groups
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
Full Text: DOI arXiv Euclid


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