The stable derivation algebras for higher genera (Q1424110)

Let \(C\) be a non-singular curve of genus \(g\geq 1\) with one point punctured over the field \({\mathbb Q}\) of rational numbers. For a fixed prime number \(\ell\), there is a Galois representation \[ \varphi_C^{\text{pro-}\ell}:\text{Gal}(\overline {\mathbb Q}/{\mathbb Q})\to \text{Out}\, \pi_1^{\text{pro-}\ell}(\overline C) =\Aut\pi_1^{\text{pro-}\ell}(\overline C)/\text{Int}\, \pi_1^{\text{pro-}\ell}(\overline C) \] attached to the pro-\(\ell\) fundamental group \(\pi_1^{\text{pro-}\ell}(\overline C)\) where \(\overline C=C\times_{\mathbb Q}\overline {\mathbb Q}\). The main concern of this paper is the image of \(\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})\) under \(\varphi_C^{\text{pro-}\ell}\). Let \[ C^{(r)}=\underbrace{C\times\cdots \times C}_{r}\,\setminus\, (\text{weak diagonal}) =\{(P_1,\dots, P_r)\mid P_i\in C, \;P_i\neq P_j\;(i\neq j)\,\} \] denote the configuration space of ordered \(r\) points on \(C\), and let \[ \cdots \longrightarrow C^{(r)}@>{p_r^{(r)}}>> C^{(r-1)} \cdots \longrightarrow C^{(2)}@>{p_r^{(2)}}>> C^{(1)}=C \] be a sequence of projections, where \(p_r^{(r)}: C^{(r)}\to C^{(r-1)}\) is defined by forgetting the \(r\)-th point. Further let \[ \varphi_{C^{(r)}}^{\text{pro-}\ell}:\text{Gal}(\overline {\mathbb Q}/{\mathbb Q})\to\text{Out}\,\pi_1^{\text{pro-}\ell}(\overline C^{(r)}) \] be a system of Galois representations compatible with the sequence of projections. Let\break \(\text{Out}^{\flat}\,\pi_1^{\text{pro-}\ell}(\overline C^{(r)})\) be certain subgroups of \(\text{Out}\,\pi_1^{\text{pro-}\ell}(\overline C^{(r)})\) which contain the image of \(\text{Gal}(\overline {\mathbb Q}/{\mathbb Q})\) and which admit homomorphisms \[ \psi_r^{(r)}: \text{Out}^{\flat}\,\pi_1^{\text{pro-}\ell}(\overline C^{(r)})\to \text{Out}^{\flat}\,\pi_1^{\text{pro-}\ell}(\overline C^{(r-1)}) \] induced from \(p_r^{(r)}\). This paper considers graded Lie algebraization of this representation. There is a weight filtration in the fundamental group \(\pi_1^{\text{pro-}\ell}(\overline C)\), which induces a filtration in \(\text{Out}\,\pi_1^{\text{pro-}\ell}(\overline C)\) and then a filtration in \(\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})\) via \(\varphi_C^{\text{pro-}\ell}\). Taking the associated graded Lie algebra structures, one obtains graded Lie algebraization \(\varphi_C^{\text{Lie}}\) of \(\varphi_C^{\text{pro-}\ell}\): \[ \varphi_C^{\text{Lie}}: {\mathcal{G}}_C\to \text{Out}^{\flat}\,\text{Gr}\, \Pi_{g,1} \] where \({\mathcal{G}}_C\) is a graded Lie algebra associated to \(\pi_1^{\text{pro-}\ell}(\overline C)\) and \(\Pi_{g,1}\) is a graded Lie algebras associated to a ``universal'' exterior monodromy representation of \(\pi_1({\mathcal{M}}_{g,1})\) on \(\pi_1\). (For the precise definition of \(\text{Gr}\,\Pi_{g,1}\), see \textit{H. Nakamura}, \textit{N. Takao} and \textit{R. Ueno} [Math. Ann. 302, 197--213 (1995; Zbl 0826.14016)].) Further consider a system of homomorphisms of graded Lie algebras \[ \varphi_{C^{(r)}}^{\text{Lie}}: \mathcal{G}_C\to \text{Out}^{\flat}\,\text{Gr}\,\Pi_{g,1}^{(r)} \] whose images are preserved in a sequence of graded Lie algebras \[ \begin{multlined} \cdots\longrightarrow \text{Out}^{\flat}\, \text{Gr}\, \Pi_{g,1}^{(r)} @>{\psi_r^{(r)}}>> \text{Out}^{\flat}\,\text{Gr}\,\Pi_{g,1}^{(r-1)}\cdots \longrightarrow \text{Out}^{\flat}\,\text{Gr}\,\Pi_{g,1}^{(2)} @>{\psi_2^{(2)}}>> \text{Out}^{\flat}\, \text{Gr}\,\Pi_{g,1}^{(1)}\\ =\text{Out}^{\flat} \,\text{Gr}\,\Pi_{g,1}.\end{multlined} \] (Here for a Lie algebra \(\mathcal{L}\), \(\text{Out}\mathcal{L}\) denotes the outer derivation algebra of \(\mathcal{L}\), and \(\text{Out}^{\flat}\mathcal{L}\) a certain subalgebra.) A special interest here is the so-called stable derivation algebra for genus \(g\), which is a certain subalgebra of \(\text{Out}^{\flat}\,\text{Gr}\,\Pi_{g,1}\). Theorem 1: For any \(g\geq 1\) and \(r\geq 4\), the homomorphism \[ \psi_r^{(r)}: \text{Out}^{\flat}\text Gr\,\Pi_{g,1}^{(r)}\to \text{Out}^{\flat}\,\text{Gr}\, \Pi_{g,1}^{(r-1)} \] is surjective (and hence bijective). Previously, it was known by \textit{Y. Ihara} and \textit{M. Kaneko} [Osaka J. Math. 29, 1--19 (1992; Zbl 0787.14015)] and Nakamura-Takao-Ueno [loc. cit.] that for every \(r\), this homomorphism is injective. Theorem 1 is actually proved working on a graded Lie algebra \({\mathcal{L}}_g^{(r)}\) over \({\mathbb Z}\) satisfying that \[ {\mathcal{L}}_g^{(r)}\otimes_{\mathbb Z}{\mathbb Z}_{\ell}\simeq \text{Gr}\,\Pi_{g,1}^{(r)}, \] and a suitable subalgebra \({\mathcal{D}}_g^{(r)}\) of the derivation algebra \(\text{Der}{\mathcal{L}}_g^{(r)}\) satisfying that \[ {\mathcal{D}}_g^{(r)}\otimes_{\mathbb Z}{\mathbb Z}_{\ell}\simeq \text{Out}^{\flat}\,\text{Gr}\,\Pi_{g,1}^{(r)}. \] Theorem 2: For any \(g\geq 1\) and \(r\geq 4\), the homomorphism \[ \psi_r^{(r)}: {\mathcal{D}}_g^{(r)}\to {\mathcal{D}}_g^{(r-1)} \] is bijective. The property established in the above theorems is called the stability property. For \(g=0\), this property was established by \textit{Y. Ihara} [Isr. J. Math. 80, 135--153 (1992; Zbl 0782.20034)]. The main result of this paper generalizes Ihara's result to curves of positive genera.