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Abstract
We prove a generalization of Tukia's ('85) isomorphism theorem which states that isomorphisms between geometrically finite groups extend equivariantly to the boundary. 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 typepreserving isomorphisms of geometrically finite subgroups of finitedimensional rank one symmetric spaces of noncompact type (ROSSONCTs) whose boundary extensions are not quasisymmetric. A sufficient condition for a typepreserving 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 finitedimensional ROSSONCTs.
Original language  English 

Pages (fromto)  659680 
Number of pages  21 
Journal  Annales academiae scientiarum fennicae series a1Mathematica 
Volume  41 
DOIs  
Publication status  Published  31 Dec 2016 
Bibliographical note
This paper is split off from an older version (v5) of arXiv:1409.2155Keywords
 math.DS
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research