Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 41, 2016, 659-680

# TUKIA'S ISOMORPHISM THEOREM IN CAT(–1) SPACES

## Tushar Das, David Simmons and Mariusz Urbanski

University of Wisconsin-La Crosse, Department of Mathematics

La Crosse, WI 54601, U.S.A.; tdas 'at' uwlax.edu

University of York, Department of Mathematics

Heslington, York YO10 5DD, U.K.; david.simmons 'at' york.ac.uk

University of North Texas, Department of Mathematics

1155 Union Circle 311430, Denton, TX 76203-5017, U.S.A.;
urbanski 'at' unt.edu

**Abstract.**
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.

**2010 Mathematics Subject Classification:**
Primary 20F69, 53C24; Secondary 20F67, 20H10, 30F40.

**Key words:**
CAT(–1) spaces, rank one symmetric spaces of
noncompact type, Kleinian groups, rigidity theorems, geometrically finite groups,
Hausdorff dimension, quasisymmetric maps.

**Reference to this article:** T. Das, D. Simmons and M. Urbanski:
Tukia's isomorphism theorem in CAT(–1) spaces.
Ann. Acad. Sci. Fenn. Math. 41 (2016), 659-680.

Full document as PDF file

doi:10.5186/aasfm.2016.4141

Copyright © 2016 by Academia Scientiarum Fennica