A quantitative version of the Morse lemma and ideal boundary fixing quasiisometries

Vladimir Shchur (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

The article is devoted to a proof of the optimal upper-bound for Morse Lemma, its "anti"-version and their applications. Roughly speaking, Morse Lemma states that in a hyperbolic metric space, a $\lambda$-quasi-geodesic $\gamma$ sits in a $\lambda^2$-neighborhood of every geodesic $\sigma$ with same endpoints. Anti-Morse Lemma states that $\sigma$ sits in a $\log\lambda$-neighborhood of $\gamma$. Applications include the displacement of points under quasi-isometries fixing the ideal boundary.

Mots clés: Morse lemma; quasi-isometries; quasi-geodesics; hyperbolic spaces; hyperbolic groups

Domaines: Géométrie métrique [math.MG]

Vladimir Shchur. A quantitative version of the Morse lemma and ideal boundary fixing quasiisometries. Journal of Functional Analysis, 2013, 264 (3), pp.815-836. ⟨10.1016/j.jfa.2012.11.014⟩. ⟨hal-00636904v2⟩