Thurston-Bonahon's theorem

The Thurston-Bonahon Theorem is a frequently used theorem from the mathematical field of 3-dimensional topology, named after William Thurston and Francis Bonahon . He specifies the dichotomy between geometrically finite and geometrically infinite surfaces in hyperbolic 3-manifolds.

Formulation of the sentence

Let it be a hyperbolic 3-manifold of finite volume and let it be an incompressible, -incompressible surface . ${\ displaystyle M}$ ${\ displaystyle F \ subset M}$ ${\ displaystyle \ partial}$

Then either a virtual fiber or quasi-fox-like . ${\ displaystyle F}$

Explanations:

${\ displaystyle F \ subset M}$ is called geometrically finite if the image of under is a geometrically finite group ; in the case of surface groups this is equivalent to being a quasi-Fuchs group . ${\ displaystyle \ pi _ {1} F}$ ${\ displaystyle \ pi _ {1} M \ to Isom (H ^ {3})}$ ${\ displaystyle \ pi _ {1} F}$

${\ displaystyle F \ subset M}$ is called virtual fiber if there is a finite superposition and a fiber bundle with fibers . In particular, Thurston-Bonahon's theorem says that every geometrically infinite surface in a hyperbolic 3-manifold of finite volume must be a virtual fiber. ${\ displaystyle p \ colon {\ widehat {M}} \ to M}$ ${\ displaystyle {\ widehat {M}} \ to S ^ {1}}$ ${\ displaystyle p ^ {- 1} (F)}$

history

Thurston-Bonahon's theorem results from a combination of sentences in Thurston's "Lecture Notes" and Bonahon's habilitation thesis with earlier results by Albert Marden . It is not explicitly mentioned either in Thurston or in Bonahon.

The theorem is used in numerous mathematical works on the topology of surfaces in 3-manifolds, explicit formulations of the theorem can be found first in Cooper-Long-Reid and in a more general form in Canary .

Individual evidence

^ William P. Thurston : The Geometry and Topology of Three Manifolds. Lecture Notes. Princeton University, Princeton NJ 1976-1979, ( online ).

^ Francis Bonahon : Bouts des variétés hyperboliques de dimension 3. In: Annals of Mathematics . Series 2, Vol. 124, No. 1, 1986, pp. 71-158, doi : 10.2307 / 1971388 .

^ Albert Marden : The geometry of finitely generated Kleinian groups. In: Annals of Mathematics. Series 2, Vol. 99, No. 3, 1974, pp. 383-762, doi : 10.2307 / 1971059 .

^ Theorem 1.1 in: Daryl Cooper, Darren D. Long, Alan W. Reid: Bundles and finite foliations. In: Inventiones Mathematicae . Vol. 118, No. 2, 1994, pp. 255-283, doi : 10.1007 / BF01231534 .

^ Corollary 8.3 in: Richard D. Canary: A covering theorem for hyperbolic 3-manifolds and its applications. In: Topology. Vol. 35, No. 3, 1996, pp. 751-778, ( digital version (PDF; 2.5 MB) ).

[1] William P. Thurston : The Geometry and Topology of Three Manifolds. Lecture Notes. Princeton University, Princeton NJ 1976-1979, ( online ).

[2] Francis Bonahon : Bouts des variétés hyperboliques de dimension 3. In: Annals of Mathematics . Series 2, Vol. 124, No. 1, 1986, pp. 71-158, doi : 10.2307 / 1971388 .

[3] Albert Marden : The geometry of finitely generated Kleinian groups. In: Annals of Mathematics. Series 2, Vol. 99, No. 3, 1974, pp. 383-762, doi : 10.2307 / 1971059 .

[4] Theorem 1.1 in: Daryl Cooper, Darren D. Long, Alan W. Reid: Bundles and finite foliations. In: Inventiones Mathematicae . Vol. 118, No. 2, 1994, pp. 255-283, doi : 10.1007 / BF01231534 .