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.
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Dehn showed that the Teichmüller modular group of a compact oriented surface is finitely generated, a set of generators being given by some Dehn twists. McCool (1975) showed that it is finitely presented.
The Teichmüller modular group of a torus is the modular group 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.
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 group Aut(π1(S,p))/π1(S,p) of π1(S,p). The Dehn–Nielsen theorem (Nielsen 1927) 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.
The Teichmüller modular groups act 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 Teichmüller modular group behaves in some ways like the automorphism group of a free group. 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. Ivanov (1992) 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.