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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
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##### References:
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