Nielsen-Thurston classification

In mathematics , the Nielsen-Thurston classification describes the possible types of self-mapping of surfaces .

Building on the work of Jakob Nielsen , it was proven in 1976 by William Thurston by means of the compactification of the Teichmüller area he constructed . Lipman Bers gave a direct proof using Teichmüller's theory .

classification

Be a closed, orientable surface of gender and be ${\ displaystyle S}$ ${\ displaystyle g \ geq 2}$

{\ displaystyle f \ colon S \ to S}

an orientation preserving homeomorphism . Then at least one of the following three alternatives applies . ${\ displaystyle f}$

${\ displaystyle f}$ is periodically : there is a so -isotopic to identity mapping is ${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle f ^ {n}}$

{\ displaystyle f}

is reducible : there is a finite family of disjoint simple closed curves which are permuted to isotopic by

{\ displaystyle f}

{\ displaystyle f}

is pseudo-Anosovian , i.e. H. Isotope to a pseudo-Anosov diffeomorphism

Proof idea

Thurston constructed a compactification of the Teichmuller space of the surface through the production space of the measured laminations on , so that the effect of a homeomorphism on this compactification is continuous. Thurston's compactification is homeomorphic to the closed (6g-6) -dimensional full sphere. According to Brouwer's fixed point theorem , the effect of must therefore have a fixed point. There are then the following options: ${\ displaystyle S}$ ${\ displaystyle S}$ ${\ displaystyle f}$

if the effect of has a fixed point inside, i.e. in the Teichmüller space, then it is periodic and the fixed point corresponds to a hyperbolic metric, with respect to which is isotopic to an isometry ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle f}$
if it is reducible, i.e. a multicurve fixed load up to isotopy, then the effect of has a fixed point in the edge of the compacting of the pond miller area ${\ displaystyle f}$ ${\ displaystyle f}$
if the effect of two fixed points at the edge of the compactification of the Teichmüller space is pseudo-Anosovian and the two fixed points correspond to the stable and unstable lamination of the too isotopic pseudo-Anosov diffeomorphism ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle f}$

Geometrization of illustration tori

Thurston used the classification of the homeomorphisms of surfaces to prove the geometrization of 3-dimensional mapping torques. This is as follows:

if is periodic, then the mapping torus has -geometry

{\ displaystyle f}

{\ displaystyle \ mathbb {H} ^ {2} \ times \ mathbb {R}}

if is reducible, then the mapping torus has a nontrivial JSJ decomposition ${\ displaystyle f}$
if is pseudo-Anosov, then the mapping torus has a hyperbolic structure ${\ displaystyle f}$

algorithm

There are numerous algorithms that enable the determination of the Nielsen-Thurston type of a mapping class in polynomial time (with regard to the word length in the mapping class group ).

literature

J. Nielsen: Surface transformation classes of algebraically finite type , Danske Vid. Selsk. Math.-Phys. Medd. 21, 89 (1944)
W. Thurston: On the geometry and dynamics of diffeomorphisms of surfaces , Bull. Amer. Math. Soc. 19, 417-431 (1988)
L. Bers: An extremal problem for quasiconformal mappings and a theorem by Thurston , Acta Math. 141, 73-98 (1978)
A. Fathi, F. Lauterbach, V. Poenaru: Travaux de Thurston , Asterisque 66/67 (1979)