Algebraic properties of mapping class groups of surfaces.

*(English)*Zbl 0635.57004
Geometric and algebraic topology, Banach Cent. Publ. 18, 15-35 (1986).

[For the entire collection see Zbl 0626.00024.]

This is a very nice survey article about recent results on the algebraic structure of the mapping class groups of compact orientable surfaces X, due to the author and others. “To expose the matter in perspective, many other results are mentioned. The paper is addressed to non-experts, and therefore we have included necessary preliminaries from the topology of surfaces and the theory of Teichmüller spaces. Proofs are omitted, but motivations, ideas of proofs and technical tools will be explained in detail.”

The topics treated are the following (where \(Mod_ X\) \(=\) modular group of X \(\cong\) mapping class group of X): Teichmüller spaces; Cohomological dimension and finiteness properties, Cohomology of \(Mod_ X\); The Thurston boundary of the Teichmüller space and the classification of elements of \(Mod_ X\); Subgroups of \(Mod_ X\); Automorphisms of \(Mod_ X\) and related groups; \(Mod_ X\) and arithmetic groups.

This is a very nice survey article about recent results on the algebraic structure of the mapping class groups of compact orientable surfaces X, due to the author and others. “To expose the matter in perspective, many other results are mentioned. The paper is addressed to non-experts, and therefore we have included necessary preliminaries from the topology of surfaces and the theory of Teichmüller spaces. Proofs are omitted, but motivations, ideas of proofs and technical tools will be explained in detail.”

The topics treated are the following (where \(Mod_ X\) \(=\) modular group of X \(\cong\) mapping class group of X): Teichmüller spaces; Cohomological dimension and finiteness properties, Cohomology of \(Mod_ X\); The Thurston boundary of the Teichmüller space and the classification of elements of \(Mod_ X\); Subgroups of \(Mod_ X\); Automorphisms of \(Mod_ X\) and related groups; \(Mod_ X\) and arithmetic groups.

Reviewer: B.Zimmermann

##### MSC:

57M99 | General low-dimensional topology |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

30F20 | Classification theory of Riemann surfaces |