2011/11/30 by John M. Mackay, Alessandro Sisto · 13 citations
Mathematics · #Geology #Geometric and Algebraic Topology #Geometry #Hyperbolic function #Hyperbolic manifold #Hyperbolic triangle #Mathematics #Mathematics and Applications #Point processes and geometric inequalities #Pure mathematics #Relatively hyperbolic group #math.GR #math.GT #math.MG #msc:20F65 #msc:20F67 #msc:51F99
paper · pdf · doi:10.5186/aasfm.2020.4511
published in Annales Academiae Scientiarum Fennicae Mathematica 45(1), 139-174 (Finnish Academy of Science and Letters) · v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4 figures. An application (Theorem 1.2) is weakened as there was an error in its proof in section 7, all other changes minor, improved exposition
arxiv created 2018/09/14 · openalex created_date 2019/08/22 · openalex publication_date 2020/01/01 · arxiv updated 2020/11/09 · openalex updated_date 2026/08/05
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.