Flows, Fixed Points and Rigidity for Kleinian Groups
| dc.creator | Biswas, Kingshook | |
| dc.date | 2009-03-13 | |
| dc.date.accessioned | 2026-07-07T12:52:25Z | |
| dc.date.available | 2026-07-07T12:52:25Z | |
| dc.description | We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group $G$, by a $G$-invariant pattern we mean a $G$-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of $G$. We prove that (in dimension at least three) for $G_1, G_2$ cocompact Kleinian groups, any quasiconformal map pairing a $G_1$-invariant pattern to a $G_2$-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas-Mj who proved rigidity for Poincare Duality subgroups. | |
| dc.identifier | https://arxiv.org/abs/0903.2419 | |
| dc.identifier | http://arxiv.org/abs/0903.2419 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/223289 | |
| dc.subject | Geometric Topology | |
| dc.subject | Dynamical Systems | |
| dc.subject | 57M50 | |
| dc.title | Flows, Fixed Points and Rigidity for Kleinian Groups | |
| dc.type | text |