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A note on the normal subgroups of mapping class groups. (English) Zbl 0584.57008
This paper is motivated by the question of whether mapping class or braid groups (denote either of these by M) can be linear groups. Several people have noticed that the nonexistence of a normal subgroup in M all of whose nonidentity elements are pseudo-Anosov proves the faithfulness of the Burau representation. However such subgroups are shown to exist in \(SL_ 2({\mathbb{Z}})\) and in the 3-strand braid group.
It is also shown that two noncentral normal subgroups of M must have nontrivial intersection and this is used to derive information about the reducible representations of M, including re-proofs of some results of Birman.
Using a theorem of Ivanov, the paper shows that finitely generated subgroups of M have torsion Frattini subgroups; in particular: (a) All braid groups have trivial Frattini subgroup, and (b) The Frattini subgroup of the mapping class group is \({\mathbb{Z}}_ 2\) in genus two, and trivial otherwise. These results are motivated by a theorem of Platonov, which says that a finitely generated linear group has nilpotent Frattini subgroup.

MSC:
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F36 Braid groups; Artin groups
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References:
[1] Platonov, Soviet Math. Dokl 7 pp 1557– (1966)
[2] DOI: 10.1002/cpa.3160220508 · Zbl 0184.49001
[3] Lyndon, Combinatorial Group Theory (1977)
[4] Birman, Braids, Links and Mapping Class Groups (1974)
[5] Fathi, Ast?risque 66?67 pp 5– (1979)
[6] Casson, Automorphisms of Surfaces after Nielsen and Thurston (1982)
[7] DOI: 10.1112/jlms/s2-9.1.160 · Zbl 0292.20032
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