Normal all pseudo-Anosov subgroups of mapping class groups.

*(English)*Zbl 0962.57007Thurston’s classification theorem: Any homeomorphism of a closed oriented surface with negative Euler characteristic is homotopic to one of the following: (i) a finite-periodic homeomorphism (elliptic type, also called finite order element), (ii) a homeomorphism which leaves invariant an essential closed 1-submanifold (parabolic type, also called reducible element), (iii) a homeomorphism which leaves invariant two singular foliations with transverse measures and stretching one and shrinking another (hyperbolic type, also called pseudo-Anosov element). W. P. Thurston’s result announced in [Bull. Am. Math. Soc., New Ser. 19, No. 2, 417-431 (1988; Zbl 0674.57008)] [see also the papers of A. Fathi, F. Laudenbach, V. Poenaru et al., Astérisque 66-67 (1979; Zbl 0446.57005 ff)] gives a characterization of elements in the mapping class group by their dynamic behavior. See [Y.-Q. Wu, Acta Math. Sin., New Ser. 3, No. 4, 305-313 (1987; Zbl 0651.57008)] for Thurston’s classification theorem for nonorientable surfaces.

Studying prescribed subgroups in the mapping class group is one of the main methods. For instance, Birman, Lubotzky and McCarthy analyzed that any abelian subgroup is finitely generated with torsion-free rank bounded above; and McCarthy showed that any subgroup either contains an abelian subgroup of finite index or a nonabelian free group.

The paper under review studies the question if pseudo-Anosov elements can form a nontrivial normal subgroup of the mapping class group. The method is to construct a normal subgroup which doesn’t contain any finite order and reducible elements. The construction starts from “Brunnian” mapping classes of a sphere braid which is analogous to a Brunnian link (every proper sublink is unlinked). Then the author shows that every nontrivial element in the Brunnian subgroup \(Br(S^2,n)\) of a mapping class group of the \(n\)-punctured sphere for \(n\geq 5\) is pseudo-Anosov. By Birman and Hilden with a 2-fold branched covering of \(\Sigma_2\to S^2\) along 6 points (where \(\Gamma_2\) is a closed orientable surface of genus 2), one has \(p:M (\Sigma_2)\to M(S^2,6)\) and lifts \(Br(S^2,6)\) to \(Br(\Sigma_2)= p^{-1}(Br (S^2,6))\). Note that the group of covering transformations of \(\Sigma_2\) is generated by a hyperelliptic involution \(i\) which is a central element in \(M(\Sigma_2)\). The involution in \(Br(\Sigma_2)\) is not pseudo-Anosov. Note that the intersection of two nontrivial noncentral normal subgroups of a mapping class group is nontrivial by a result of Long. Taking a representation \(\rho:M (\Sigma_2)\to Sp(4,Z_3)\) to kill the involution, one has \(Br(\Sigma_2) \cap\text{ker} \rho\) nontrivial since both \(Br(\Sigma_2)\) and \(\text{ker} \rho\) are not central. Therefore one obtains a nontrivial normal subgroup (of the mapping class group of a closed oriented genus 2 surface) which consists of only pseudo-Anosov elements.

Studying prescribed subgroups in the mapping class group is one of the main methods. For instance, Birman, Lubotzky and McCarthy analyzed that any abelian subgroup is finitely generated with torsion-free rank bounded above; and McCarthy showed that any subgroup either contains an abelian subgroup of finite index or a nonabelian free group.

The paper under review studies the question if pseudo-Anosov elements can form a nontrivial normal subgroup of the mapping class group. The method is to construct a normal subgroup which doesn’t contain any finite order and reducible elements. The construction starts from “Brunnian” mapping classes of a sphere braid which is analogous to a Brunnian link (every proper sublink is unlinked). Then the author shows that every nontrivial element in the Brunnian subgroup \(Br(S^2,n)\) of a mapping class group of the \(n\)-punctured sphere for \(n\geq 5\) is pseudo-Anosov. By Birman and Hilden with a 2-fold branched covering of \(\Sigma_2\to S^2\) along 6 points (where \(\Gamma_2\) is a closed orientable surface of genus 2), one has \(p:M (\Sigma_2)\to M(S^2,6)\) and lifts \(Br(S^2,6)\) to \(Br(\Sigma_2)= p^{-1}(Br (S^2,6))\). Note that the group of covering transformations of \(\Sigma_2\) is generated by a hyperelliptic involution \(i\) which is a central element in \(M(\Sigma_2)\). The involution in \(Br(\Sigma_2)\) is not pseudo-Anosov. Note that the intersection of two nontrivial noncentral normal subgroups of a mapping class group is nontrivial by a result of Long. Taking a representation \(\rho:M (\Sigma_2)\to Sp(4,Z_3)\) to kill the involution, one has \(Br(\Sigma_2) \cap\text{ker} \rho\) nontrivial since both \(Br(\Sigma_2)\) and \(\text{ker} \rho\) are not central. Therefore one obtains a nontrivial normal subgroup (of the mapping class group of a closed oriented genus 2 surface) which consists of only pseudo-Anosov elements.

Reviewer: Weiping Li (Stillwater)

##### MSC:

57M60 | Group actions on manifolds and cell complexes in low dimensions |

20F36 | Braid groups; Artin groups |

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

**OpenURL**

##### References:

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