Relative hyperbolicity of hyperbolic-by-cyclic groups

Groups, geometry, and dynamics

article

François Dahmani¹  Suraj Krishna M S² 

[1] Université Grenoble Alpes;Tata Institute of Fundamental Research;Technion – Israel Institute of Technology

关键词: Automorphisms of groups;    relative hyperbolicity;    semidirect products;   

DOI  :  10.4171/ggd/703

学科分类：神经科学

来源: European Mathematical Society

【 摘 要 】

Let GGG be a torsion-free hyperbolic group and α\alphaα an automorphism of GGG. We show that there exists a canonical collection of subgroups that are polynomially growing under α\alphaα, and that the mapping torus of GGG by α\alphaα is hyperbolic relative to the suspensions of the maximal polynomially growing subgroups under α\alphaα. 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.

【 授权许可】

CC BY