Indistinguishability of conjugacy classes of the pro-l mapping class group.

*(English)*Zbl 0705.20037From the introduction: “Let \(\ell\) be a fixed prime number and \(\pi^{(g)}\) denote the pro-\(\ell\) completion of the topological fundamental group of a compact Riemann surface of genus \(g\geq 2\). So, we have \(\pi^{(g)}=F/N\), where F is the free pro-\(\ell\) group of rank 2g generated by \(x_ 1,...,x_{2g}\) and N is the closed normal subgroup of F which is normally generated by \([x_ 1,x_{g+1}]...[x_ g,x_{2g}]\), [, ] being the commutator: \([x,y]=xyx^{-1}y^{-1}\) (x,y\(\in F)\). We denote by \(\Gamma_ g\) the outer automorphism group of \(\pi^{(g)}\) and call it the pro-\(\ell\) mapping class group. Let \(\lambda\) : \(\Gamma\) \({}_ g\to GSp(2g,Z_{\ell})\) be the canonical homomorphism induced by the action of \(\Gamma_ g\) on \(\pi^{(g)}/[\pi^{(g)},\pi^{(g)}]\). We treat the case \(g=2\). Then, our result is the following Theorem: Assume that \(\ell \geq 5\). Then, there exists an integer \(N\geq 1\) such that the following statement holds: If \(A\in GSp(4,Z_{\ell})\) satisfies the condition \(A\equiv \ell_ 4 mod \ell^ N\), \(\lambda^{-1}(C_ A)\) contains more than one \(\Gamma_ 2\)-conjugacy class. Here, \(C_ A\) denotes the \(GSp(4,Z_{\ell})\)-conjugacy class containing A. In a previous paper, we have proved this “indistinguishability of conjugacy class” under the assumption that \(g\geq 3\).... So, to prove the above theorem, we use the method “calculations modulo \(\pi^{(g)}(4)''\). Although this requires rather complicated calculations, it is carried out by using the “Lie algebra” of the nilpotent pro-\(\ell\) group \(\pi^{(g)}/\pi^{(g)}(4)\).”

Reviewer: T.Nôno

##### MSC:

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

20E18 | Limits, profinite groups |

30F10 | Compact Riemann surfaces and uniformization |

20F28 | Automorphism groups of groups |

20F40 | Associated Lie structures for groups |

20F14 | Derived series, central series, and generalizations for groups |

##### Keywords:

pro-\(\ell \) completion; topological fundamental group; compact Riemann surface; free pro-\(\ell \) group; outer automorphism group; pro-\(\ell \) mapping class group; conjugacy class; Lie algebra; nilpotent pro-\(\ell \) group
PDF
BibTeX
Cite

\textit{M. Asada}, Proc. Japan Acad., Ser. A 64, No. 7, 256--259 (1988; Zbl 0705.20037)

Full Text:
DOI

##### References:

[1] | M. Asada: An analogue of the Levi decomposition of the automorphism group of certain nilpotent pro-J group (1988) (preprint). · Zbl 0705.20035 |

[2] | M. Asada and M. Kaneko: On the automorphism group of some pro4 fundamental groups. Advanced Studies in Pure Math., vol. 12, pp. 137-159 (1987). · Zbl 0657.20028 |

[3] | Y. Ihara: Profinite braid groups, Galois representations and complex multiplications. Ann. of Math., 123, 43-106 (1986). JSTOR: · Zbl 0595.12003 |

[4] | Y. Ihara: Some problems on three point ramifications and associated large Galois representations. Advanced Studies in Pure Math., vol. 12, pp. 173-188 (1987). · Zbl 0659.12014 |

[5] | M. Kaneko: On conjugacy classes of the pro4 braid group of degree 2. Proc. Japan Acad., 62A, 274-277 (1986). · Zbl 0618.20022 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.