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**Indistinguishability of conjugacy classes of the pro-l mapping class group.**
*(English)*
Zbl 0705.20037

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

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### 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 |

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