The center of the Goldman Lie algebra of a surface of infinite genus (Q2871304)

Let \(\Sigma\) be a connected oriented surface and let \(\hat \pi\) be the set of conjugacy classes in the fundamental group \(\pi=\pi_1(\Sigma)\). W.~Goldman introduced in the 80's a Lie bracket on the \(\mathbb{Q}\)-vector space \(\mathbb{Q} \hat \pi\) generated by the set \(\hat \pi\). In 2006, P. Etingof proved the following result: if the surface \(\Sigma\) is closed, then the center \(Z(\mathbb{Q} \hat \pi)\) of the Goldman Lie algebra \(\mathbb{Q} \hat \pi\) is \textit{trivial}, in the sense that it is \(\mathbb{Q}\)-spanned by the trivial loop \(\circlearrowleft\, \in \hat \pi\).NEWLINENEWLINEThe paper under review shows that the center \(Z(\mathbb{Q} \hat \pi)\) is also trivial if \(\Sigma\) is the surface \(\Sigma_{\infty,1}\) of infinite genus with one boundary component. In the closed case, Etingof's argument was based on the original relation between the Goldman bracket and the canonical Poisson structure on the moduli space of flat \(\text{GL}_N(\mathbb{C})\)-connections. In the present case, the proof is based on a natural action by derivations of the Goldman Lie algebra \(\mathbb{Q} \hat \pi\) on (the \(I\)-adic completion of) the group algebra of \(\pi\), which the authors introduced in a previous work. When \(\Sigma\) is the compact surface \(\Sigma_{g,1}\) of genus \(g\) with one boundary component, this action can be described in an algebraic way using an appropriate ``expansion'' of the free group \(\pi\) into the degree-completion \(\widehat T\) of the tensor algebra \(T:=T(H)\) over \(H:=H_1(\Sigma;\mathbb{Q})\). Thus the authors obtain a Lie algebra map \( \mathbb{Q} \hat \pi \to \text{Der}_{\omega}(\widehat T), \) with values in the Lie algebra of derivations of \(\widehat T\) that vanish on the symplectic form \(\omega \in H \otimes H\) defined by homological intersection. Note that \(\text{Der}_{\omega}(\widehat T)\) is essentially the Lie algebra introduced by M. Kontsevich under the name \(\mathfrak{a}_g\) (\(\mathfrak{a}\) for ``associative''); the center of \(\mathfrak{a}_g\) is contained in the center of the part \(\mathfrak{a}_g^{\mathfrak{sp}}\) invariant under \(\mathfrak{sp}:=\mathfrak{sp}_{2g}(\mathbb{Q})\). In order to control \(Z(\mathfrak{a}_g^{\mathfrak{sp}})\), the authors introduce the Lie algebra \(\mathcal{C}\) of ``oriented chord diagrams'', which maps onto \(\mathfrak{a}_g^{\mathfrak{sp}}\) by elementary invariant theory. They compute the center of \(\mathcal{C}\) and, since the map \(\mathcal{C} \to \mathfrak{a}_g^{\mathfrak{sp}}\) is an isomorphism in degree less than \(g\), they are able to conclude by taking the limit \(g \to \infty\). When \(\Sigma= \Sigma_{g,1}\) is a surface of finite genus \(g\), the authors also conjecture that \(Z(\mathbb{Q} \hat \pi)\) is \(\mathbb{Q}\)-spanned by all the powers of the boundary curve.