Symmetries of tropical moduli spaces of curves

Abstract

We compute the automorphism group $\mathrm{Aut}({\Delta }_{g,n})$ for all $g,n\ge 0$ such that $3g-3+n>0$, where ${\Delta }_{g,n}\subset M_{g,n}^{\mathrm{trop}}$ is the moduli space of stable $n$-marked tropical curves of genus $g$ and volume one. In particular, we show that $\mathrm{Aut}({\Delta }_g)$ is trivial for $g\ge 2$, while $\mathrm{Aut}({\Delta }_{g,n})\cong S_n$ when $n\ge 1$ and $(g,n)\ne (0,4),(1,2)$. The space ${\Delta }_{g,n}$ is a symmetric $\Delta$-complex in the sense of Chan, Galatius, and Payne, and is identified with the dual intersection complex of the boundary divisor in the Deligne-Mumford-Knudsen moduli space ${\overline{\mathcal{M}}}_{g,n}$ of stable curves. After the work of Massarenti, who has shown that $\mathrm{Aut}({\overline{\mathcal{M}}}_g)$ is trivial for $g\ge 2$ while $\mathrm{Aut}({\overline{\mathcal{M}}}_{g,n})\cong S_n$ when $n\ge 1$ and $2g-2+n\ge 3$, our result implies that the tropical moduli space ${\Delta }_{g,n}$ faithfully reflects the symmetries of the algebraic moduli space for general $g$ and $n$.

accepted version; 38 pages, 17 figures