##
**Mapping class groups of surfaces.**
*(English)*
Zbl 0663.57008

Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 13-43 (1988).

[For the entire collection see Zbl 0651.00010.]

The author surveys some aspects of the study of the mapping class groups (the group of orientation-preserving diffeomorphisms, modulo diffeotopy) of 2-dimensional surfaces. The topics covered include generators and relations, the Torelli group (the subgroup consisting of the mapping classes that induce the identity automorphism on the homology of the surface), the Thurston-Nielsen classification of diffeomorphisms, the Nielsen realization problem, and matrix representations of mapping class groups. An extensive bibliography is included.

The author provides at the outset a list of other survey articles for this large subject. The first-listed of these, J. Harer’s extensive and very well-written survey of the cohomological aspects of mapping class groups, is missing from the bibliography - it appeared in Theory of moduli [Lect. Notes Math. 1337, 138-221 (1988)]. A few oversights must be mentioned. Near the bottom of p. 14, one should note that \(g=1\) is an exceptional case. The reference to Ness on p. 23 should be Mess, and [Ne] does not appear in the bibliography (at last word, details of the work of Geoffrey Mess cited in the article will be available soon). On p. 24, the question of whether the subgroup generated by Dehn twists about simple closed curves that separate the surface is free and infinitely generated for higher genera, as it is for genus 2, is said to be open. To the reviewer’s knowledge, the question of infinite generation is open, but for higher genera these groups are certainly not free, as twists about disjoint loops which each separate will commute, and therefore the centralizers of elements will be too large for the group to be free. The definition of pseudo-Anosov diffeomorphism given on p. 27 should state that no power of the diffeomorphism fixes any nonperipheral conjugacy class in \(\pi_ 1(S_ g)\). In the summary on p. 34 of Kerckhoff’s proof of the Nielsen Realization Conjecture, the measures are measures on transverse arcs to the lamination, not measures on the entire lamination space as seems to be said there, and then Kerckhoff’s Theorem asserts that a finite subgroup of the mapping class group has a fixed point in the interior of Teichmüller space, not in the Thurston boundary.

The author surveys some aspects of the study of the mapping class groups (the group of orientation-preserving diffeomorphisms, modulo diffeotopy) of 2-dimensional surfaces. The topics covered include generators and relations, the Torelli group (the subgroup consisting of the mapping classes that induce the identity automorphism on the homology of the surface), the Thurston-Nielsen classification of diffeomorphisms, the Nielsen realization problem, and matrix representations of mapping class groups. An extensive bibliography is included.

The author provides at the outset a list of other survey articles for this large subject. The first-listed of these, J. Harer’s extensive and very well-written survey of the cohomological aspects of mapping class groups, is missing from the bibliography - it appeared in Theory of moduli [Lect. Notes Math. 1337, 138-221 (1988)]. A few oversights must be mentioned. Near the bottom of p. 14, one should note that \(g=1\) is an exceptional case. The reference to Ness on p. 23 should be Mess, and [Ne] does not appear in the bibliography (at last word, details of the work of Geoffrey Mess cited in the article will be available soon). On p. 24, the question of whether the subgroup generated by Dehn twists about simple closed curves that separate the surface is free and infinitely generated for higher genera, as it is for genus 2, is said to be open. To the reviewer’s knowledge, the question of infinite generation is open, but for higher genera these groups are certainly not free, as twists about disjoint loops which each separate will commute, and therefore the centralizers of elements will be too large for the group to be free. The definition of pseudo-Anosov diffeomorphism given on p. 27 should state that no power of the diffeomorphism fixes any nonperipheral conjugacy class in \(\pi_ 1(S_ g)\). In the summary on p. 34 of Kerckhoff’s proof of the Nielsen Realization Conjecture, the measures are measures on transverse arcs to the lamination, not measures on the entire lamination space as seems to be said there, and then Kerckhoff’s Theorem asserts that a finite subgroup of the mapping class group has a fixed point in the interior of Teichmüller space, not in the Thurston boundary.

Reviewer: D.McCullough

### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

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

57M05 | Fundamental group, presentations, free differential calculus |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |