2015/08/31 by Tushar Das, David Simmons, Mariusz Urbański
Mathematics · #Advanced Algebra and Geometry #Discrete mathematics #Equivariant map #Geometric and Algebraic Topology #Homeomorphism (graph theory) #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Isomorphism theorem #Mathematics #Pure mathematics #math.DS
paper · pdf · doi:10.5186/aasfm.2016.4141
published as Ann. Acad. Sci. Fenn. Math. 41 (2016), 659-680 · This paper is split off from an older version (v5) of arXiv:1409.2155
arxiv created 2015/11/28 · openalex publication_date 2016/01/01 · arxiv updated 2018/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We prove a generalization of Tukia's ('85) isomorphism theorem, which states that any isomorphism between two geometrically finite groups extends equivariantly to a quasisymmetric homeomorphism between their limit sets. Tukia worked in the setting of real hyperbolic spaces of finite dimension, and his theorem cannot be generalized as stated to the setting of CAT(-1) spaces. We exhibit examples of type-preserving isomorphisms of geometrically finite subgroups of finite-dimensional rank one symmetric spaces of noncompact type (ROSSONCTs) whose boundary extensions are not quasisymmetric. A sufficient condition for a type-preserving isomorphism to extend to a quasisymmetric equivariant homeomorphism between limit sets is that one of the groups in question is a lattice, and that the underlying base fields are the same, or if they are not the same then the base field of the space on which the lattice acts has the larger dimension. This in turn leads to a generalization of a rigidity theorem of Xie ('08) to the setting of finite-dimensional ROSSONCTs.