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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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[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
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