arXiv subjects
Publications and source records attributed to Vladimir Shchur.
There is a gap in the proof of the main theorem in the article [ShCh13a] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [ShCh13a] is correct. We also describe a computer certification of this result.
We consider a quantitative form of the quasi-isometry problem. We discuss several arguments which lead us to different results and bounds of quasi-isometric distortion: comparison of volumes, connectivity etc. Then we study the transport of Poincaré constants by quasi-isometries and we give sharp lower and upper bounds for the homotopy distortion growth for an interesting class of hyperbolic metric spaces.
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 $λ$-quasi-geodesic $γ$ sits in a $λ^2$-neighborhood of every geodesic $σ$ with same endpoints. Anti-Morse Lemma states that $σ$ sits in a $\logλ$-neighborhood of $γ$. Applications include the displacement of points under quasi-isometries fixing the ideal boundary.