Let be a torsion-free hyperbolic group and an automorphism of . We show that there exists a canonical collection of subgroups that are polynomially growing under , and that the mapping torus of by is hyperbolic relative to the suspensions of the maximal polynomially growing subgroups under . As a consequence, we obtain a dichotomy for growth: given an automorphism of a torsion-free hyperbolic group, the conjugacy class of an element either grows polynomially under the automorphism, or at least exponentially.
Cite this article
François Dahmani, Suraj Krishna M S, Relative hyperbolicity of hyperbolic-by-cyclic groups. Groups Geom. Dyn. 17 (2023), no. 2, pp. 403–426DOI 10.4171/GGD/703