Page "Mapping class group" Paragraph 24

The mapping class groups of surfaces have been heavily studied, and are called Teichmüller modular groups.

( Note the special case of MCG ( T < sup > 2 </ sup >) above.

) This is perhaps due to their strange similarity to higher rank linear groups as well as many applications, via surface bundles, in Thurston's theory of geometric three-manifolds.

For more information on this topic see the Nielsen – Thurston classification theorem and the article on Dehn twists.

Every finite group is a subgroup of the mapping class group of a closed, orientable surface, moreover one can realize any finite group as the group of isometries of some compact Riemann surface.