The mapping class group appeared in the first half of the twentieth century.
The earliest contributors were Max Dehn and Jakob Nielsen: Dehn proved finite generation of the group,[1] and Nielsen gave a classification of mapping classes and proved that all automorphisms of the fundamental group of a surface can be represented by homeomorphisms (the Dehn–Nielsen–Baer theorem).
The Dehn–Nielsen theory was reinterpreted in the mid-seventies by Thurston who gave the subject a more geometric flavour[2] and used this work to great effect in his program for the study of three-manifolds.
If we modify the definition to include all homeomorphisms we obtain the extended mapping class group
This definition can also be made in the differentiable category: if we replace all instances of "homeomorphism" above with "diffeomorphism" we obtain the same group, that is the inclusion
The image is exactly those outer automorphisms which preserve each conjugacy class in the fundamental group corresponding to a boundary component.
This is an exact sequence relating the mapping class group of surfaces with the same genus and boundary but a different number of punctures.
It is a fundamental tool which allows to use recursive arguments in the study of mapping class groups.
There is a classification of the mapping classes on a surface, originally due to Nielsen and rediscovered by Thurston, which can be stated as follows.
is either: The main content of the theorem is that a mapping class which is neither of finite order nor reducible must be pseudo-Anosov, which can be defined explicitly by dynamical properties.
For example, a random walk on the mapping class group will end on a pseudo-Anosov element with a probability tending to 1 as the number of steps grows.
is the space of marked complex (equivalently, conformal or complete hyperbolic) structures on
In particular, the Teichmüller metric can be used to establish some large-scale properties of the mapping class group, for example that the maximal quasi-isometrically embedded flats in
[9] The action extends to the Thurston boundary of Teichmüller space, and the Nielsen-Thurston classification of mapping classes can be seen in the dynamical properties of the action on Teichmüller space together with its Thurston boundary.
The action is not properly discontinuous (the stabiliser of a simple closed curve is an infinite group).
(isotopy classes of maximal systems of disjoint simple closed curves).
[12] The stabilisers of the mapping class group's action on the curve and pants complexes are quite large.
Two distinct markings are joined by an edge if they differ by an "elementary move", and the full complex is obtained by adding all possible higher-dimensional simplices.
The mapping class group is generated by the subset of Dehn twists about all simple closed curves on the surface.
The Dehn–Lickorish theorem states that it is sufficient to select a finite number of those to generate the mapping class group.
The smallest number of Dehn twists that can generate the mapping class group of a closed surface of genus
It is possible to prove that all relations between the Dehn twists in a generating set for the mapping class group can be written as combinations of a finite number among them.
There are other interesting systems of generators for the mapping class group besides Dehn twists.
This comes from the fact that the intersection number of closed curves induces a symplectic form on the first homology, which is preserved by the action of the mapping class group.
boil down to a statement about its Torelli subgroup) and applications to 3-dimensional topology and algebraic geometry.
An example of application of the Torelli subgroup is the following result: The proof proceeds first by using residual finiteness of the linear group
, and then, for any nontrivial element of the Torelli group, constructing by geometric means subgroups of finite index which does not contain it.
A bound on the order of finite subgroups can also be obtained through geometric means.
[22] Any subgroup which is not reducible (that is it does not preserve a set of isotopy class of disjoint simple closed curves) must contain a pseudo-Anosov element.
The images of these representations are contained in arithmetic groups which are not symplectic, and this allows to construct many more finite quotients of