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Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations Bauer, Ulrich Roll, Fabian Vietoris–Rips complexes persistent homology discrete Morse theory apparent pairs hyperbolicity geodesic defect Ripser Motivated by computational aspects of persistent homology for Vietoris–Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris–Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the filtration. We further show that for finite tree metrics the Vietoris–Rips complexes collapse to their corresponding subforests. We relate our result to modern computational methods by showing that these collapses are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ulrich Bauer and Fabian Roll 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.15 urn:nbn:de:0030-drops-160237 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.15 eng https://creativecommons.org/licenses/by/4.0/legalcode