2017/08/31 by Matthew Haulmark
Mathematics · #Advanced Operator Algebra Research #Boundary (topology) #Class (philosophy) #Combinatorics #Geometric and Algebraic Topology #Group (periodic table) #Homeomorphism (graph theory) #Homotopy and Cohomology in Algebraic Topology #Hyperbolic equilibrium point #Hyperbolic function #Hyperbolic manifold #Mathematical analysis #Mathematics #Order (exchange) #Pure mathematics #Relatively hyperbolic group #Type (biology) #math.GR #math.GT #msc:20F65 #msc:20F67
paper · pdf · doi:10.2140/agt.2019.19.2795
published as Algebr. Geom. Topol. 19 (2019) 2795-2836 · 35 pages
arxiv created 2018/06/14 · openalex publication_date 2019/10/20 · arxiv updated 2019/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that the existence of a nonparabolic local cut point in the Bowditch boundary [math] of a relatively hyperbolic group [math] implies that [math] splits over a [math] –ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of [math] –dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over [math] –ended subgroups and no peripheral splittings.\n¶ In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua [math] and [math] , then [math] is homeomorphic to [math] . Thus we propose an alternative definition of [math] which increases the class of spaces on which [math] can act.