Relative hyperbolicity of hyperbolic-by-cyclic groups

  • François Dahmani

    Université Grenoble Alpes, France
  • Suraj Krishna M S

    Tata Institute of Fundamental Research, Mumbai, India; Technion – Israel Institute of Technology, Haifa, Israel
Relative hyperbolicity of hyperbolic-by-cyclic groups cover
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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.

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François Dahmani, Suraj Krishna M S, Relative hyperbolicity of hyperbolic-by-cyclic groups. Groups Geom. Dyn. 17 (2023), no. 2, pp. 403–426

DOI 10.4171/GGD/703