A combinatorial construction of symplectic expansions (Q2880669)

Let \(\Sigma\) be an oriented surface of genus \(g>0\) with one boundary component, and let \(\pi\) denote its fundamental group. The associated mapping class group \(\mathcal{M}_{g,1}\) is the group of isotopy classes of orientation-preserving diffeomorphisms of \(\Sigma\) that fix the boundary pointwise. The Johnson filtration of \(\mathcal{M}_{g,1}\) is given by a nested sequence of subgroups whose \(k\)-th term \(\mathcal{M}_{g,1}[k]\), known as the \(k\)-th Torelli group, consists of those mapping classes which act trivially on the \(k\)-th nilpotent quotient of \(\pi\). For every \(k\), Johnson defined a certain abelian quotient of \(\mathcal{M}_{g,1}[k]\); these quotient maps are known as the Johnson homomorphisms, and are an important tool in the study of the algebraic structure of the mapping class group.NEWLINENEWLINEA Magnus expansion \(\Theta\) of \(\pi\) (in the sense of Kawazumi) determines an injective homomorphism of \(\mathcal{M}_{g,1}\) known as the total Johnson map associated to \(\Theta\), and induces an extension of each Johnson homomorphism to \(\mathcal{M}_{g,1}\) (though not as a homomorphism). A symplectic expansion, introduced by \textit{G. Massuyeau} [Bull. Soc. Math. Fr. 140, No. 1, 101--161 (2012; Zbl 1248.57009)], is a Magnus expansion which in some sense respects the fact that \(\pi\) has a particular element corresponding to the boundary of \(\Sigma\). As described in [\textit{N. Kawazumi} and \textit{Y. Kuno}, The logarithms of Dehn twists. (2010), \url{arXiv:1008.5017}], symplectic expansions have many nice properties; for example they are used to establish a close relationship between the Goldman Lie algebra of \(\Sigma\) and formal symplectic geometry.NEWLINENEWLINEWhile symplectic expansions had been shown to exist by Massuyeau, the construction therein starts from a symplectic generating set of \(\pi\), and involves various choices. The paper under review gives a construction which canonically associates a symplectic expansion to any free generating set of \(\pi\). The construction is purely combinatorial, and suitable for computer-aided calculation.