= Infinite cyclic subgroups of hyperbolic groups are undistorted
If $g$ has infinite order in a <hyperbolic group>, the map $n\mapsto g^n$ is a <quasi-isometric embedding> of the integers. Equivalently $|g^n|\geq c|n|-C$ for some $c>0$. This is the homogeneous quasigeodesic theorem: local-to-global control of <metric geodesics> in a hyperbolic <Cayley graph> gives a quasigeodesic orbit for a suitable conjugate of a positive power; undoing <conjugation> and passing between consecutive powers changes only uniform constants. It implies $\tau(g)>0$. The theorem and its hypotheses are given in https://loeh.app.uni-regensburg.de/teaching/ggt_ss22/lecture_notes.pdf[Löh, Theorem 6.5.9]. It is not true for arbitrary finitely generated groups, as the <Baumslag-Solitar group> relation $ab a^{-1}=b^3$ shows.
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