Teichmüller modular group
Encyclopedia
In mathematics, a Teichmüller modular group, or mapping class group of a surface, or homeotopy group of a surface, is the group of isotopy classes of orientation-preserving homeomorphisms of an oriented surface. It is also a group of automorphisms of a Teichmüller space
.
s. showed that it is finitely presented.
SL2(Z).
The Teichmüller modular group of a sphere with n points removed is the spherical braid group on n strands, which is the quotient of the Braid group
Bn−1 by its infinite cyclic center.
Aut(π1(S,p))/π1(S,p) of π1(S,p). The Dehn–Nielsen theorem states that the Teichmüller modular group is a subgroup of index 2 of this outer automorphism group, consisting of the orientation-preserving outer automorphisms, that act trivially on the second cohomology group H2(π1(S,p),Z) = H2(S,Z) = Z.
as automorphisms of the corresponding Teichmüller spaces, preserving most of the structure such as the complex structure, the Teichmüller metric, the Weil-Petersson metric, and so on. Royden proved that in the case of a compact Riemann surface of genus greater than 1, the Teichmüller modular group is the group of all biholomorphic maps of Teichmüller space.
. The reason is that the Teichmüller modular group is an index 2 subgroup of the fundamental group of a surface, and fundamental groups of surfaces are quite similar to free groups.
The Teichmüller modular group also behaves rather like a linear group. proved that it has many of the properties of linear groups. The action of the Teichmüller modular group on Teichmüller space is similar to the action of the Siegel modular group on the Siegel upper half space.
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
.
Presentation
Dehn showed that the Teichmüller modular group of a compact oriented surface is finitely generated, a set of generators being given by some Dehn twistDehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...
s. showed that it is finitely presented.
Examples
The Teichmüller modular group of a torus is the modular groupModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
SL2(Z).
The Teichmüller modular group of a sphere with n points removed is the spherical braid group on n strands, which is the quotient of the Braid group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
Bn−1 by its infinite cyclic center.
Dehn–Nielsen theorem
If S is a compact Riemann surface with basepoint p and fundamental group π1(S,p), then the group of isotopy classes of homeomorphisms of S is naturally isomorphic to the outer automorphism groupOuter automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
Aut(π1(S,p))/π1(S,p) of π1(S,p). The Dehn–Nielsen theorem states that the Teichmüller modular group is a subgroup of index 2 of this outer automorphism group, consisting of the orientation-preserving outer automorphisms, that act trivially on the second cohomology group H2(π1(S,p),Z) = H2(S,Z) = Z.
Action on Teichmüller space
The Teichmüller modular groups actGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
as automorphisms of the corresponding Teichmüller spaces, preserving most of the structure such as the complex structure, the Teichmüller metric, the Weil-Petersson metric, and so on. Royden proved that in the case of a compact Riemann surface of genus greater than 1, the Teichmüller modular group is the group of all biholomorphic maps of Teichmüller space.
Analogues with other groups
The Teichmüller modular group behaves in some ways like the automorphism group of a free groupAutomorphism group of a free group
In mathematical group theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface.-Presentation:...
. The reason is that the Teichmüller modular group is an index 2 subgroup of the fundamental group of a surface, and fundamental groups of surfaces are quite similar to free groups.
The Teichmüller modular group also behaves rather like a linear group. proved that it has many of the properties of linear groups. The action of the Teichmüller modular group on Teichmüller space is similar to the action of the Siegel modular group on the Siegel upper half space.