Symmetries of tropical moduli spaces of curves

Abstract

We compute the automorphism group $\\mathrm{Aut}(\\Delta_{g, n})$ for all $g, n \\geq 0$ 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 $g \\geq 2$, while $\\mathrm{Aut}(\\Delta_{g, n}) \\cong S_n$ when $n \\geq 1$ and $(g, n) \\neq (0, 4), (1, 2)$. 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 Masseranti, who has shown that $\\mathrm{Aut}(\\overline{\\mathcal{M}}_g)$ is trivial for $g \\geq 2$ while $\\mathrm{Aut}(\\overline{\\mathcal{M}}_{g, n}) \\cong S_n$ when $n \\geq 1$ and $2g - 2 + n \\geq 3$, our result implies that the tropical moduli space $\\Delta_{g, n}$ faithfully reflects the symmetries of the algebraic moduli space for general $g$ and $n$.