Sullivan rigidity

In mathematical sub-region of the geometrical function theory is Sullivan rigidity (engl. Sullivan rigidity ) a low-lying generalization of Mostow-rigidity in the theory of Klein groups . It goes back to Dennis Sullivan .

Sullivan's rigidity plays a central role in the proof of essential theorems of 3-dimensional topology, for example in the proof of the Ending Lamination Conjecture by Brock - Canary - Minsky and in Thurston's proof of the geometrization of fibrous 3-manifolds.

Rigidity Theorem

Let be discrete groups of isometries of hyperbolic space for which there is an isomorphism . The effect of on their Limes amount is recurrent . ${\displaystyle \Gamma _{1},\Gamma _{2}\subset \mathrm {Isom} (H^{n})}$ ${\displaystyle H^{n},n\geq 3}$ ${\displaystyle \rho \colon \Gamma _{1}\to \Gamma _{2}}$ ${\displaystyle \Gamma _{1}}$ ${\displaystyle \Lambda (\Gamma _{1})}$

If then there is a quasi-conformal homeomorphism of the boundary in the infinite of hyperbolic space such that ${\displaystyle f\colon \partial _{\infty }H^{n}\to \partial _{\infty }H^{n}}$

${\displaystyle f(\gamma (x))=\rho (\gamma )(f(x))\ \forall \gamma \in \Gamma _{1},x\in H^{n}}$

holds and the restriction from to the discontinuity region is a conformal mapping , then it must be a Möbius transformation . ${\displaystyle f}$ ${\displaystyle \Omega (\Gamma _{1})}$ ${\displaystyle f}$

Remarks

• The prerequisite for a recurrent effect on the limit set is (according to Ahlfors' theorem ) automatically fulfilled for all finitely generated groups .
• From the rigidity theorem for finitely generated Kleinian groups that each -invariant Beltrami differential with almost all be zero needs. ${\displaystyle \Gamma \subset \mathrm {Isom} ^{+}(H^{3})}$ ${\displaystyle \Gamma }$ ${\displaystyle \mu }$ ${\displaystyle \mathrm {supp} (\mu )\subset \Lambda (\Gamma )}$
• Sullivan's rigidity has numerous applications in complex dynamics and in the geometry of hyperbolic manifolds of infinite volume.

literature

• Dennis Sullivan: On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), pp. 465-496, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, NJ, 1981.
• Matsuzaki-Taniguchi: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. ISBN 0-19-850062-9 (Chapter 5.2: The Sullivan rigidity theorem )
• M. Kapovich: Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009. ISBN 978-0-8176-4912-8 (Chapter 8.6: Sullivan rigidity theorem )