an:04156775
Zbl 0705.20037
Asada, Mamoru
Indistinguishability of conjugacy classes of the pro-l mapping class group
EN
Proc. Japan Acad., Ser. A 64, No. 7, 256-259 (1988).
00170130
1988
j
20F34 20E18 30F10 20F28 20F40 20F14
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
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 \(\pi^{(g)}/\pi^{(g)}(4)\).''
T.N??no