Abstract:Put $\Delta _{g, n} \subset M_{g, n}^\mathrm {trop}$ for the moduli space of stable $n$-marked tropical curves of genus $g$ and volume one. We compute the automorphism group $Aut(\Delta _{g, n})$ for all $g, n \geq 0$ such that $3g - 3 + n > 0$. In particular, we show that $Aut(\Delta _{g})$ is trivial for $g \geq 2$, while $Aut(\Delta _{g, n}) \cong S_n$ 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 stack $\overline {\mathcal {M}}_{g, n}$ of stable curves. After the work of Massarenti [J. Lond. Math. Soc. 89 (2014), pp. 131–150], who has shown that $Aut(\overline {\mathcal {M}}_g)$ is trivial for $g \geq 2$ while $Aut(\overline {\mathcal {M}}_{g, n}) \cong S_n$ 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 $n$.

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中文翻译：

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曲线的热带模空间的对称性

摘要：将$\Delta _{g, n} \subset M_{g, n}^\mathrm {trop}$ 作为$g$属和第一卷的稳定$n$标记的热带曲线的模空间。我们为所有 $g, n \geq 0$ 计算自同构群 $Aut(\Delta _{g, n})$ 使得 $3g - 3 + n > 0$。特别地，我们证明 $Aut(\Delta _{g})$ 对于 $g \geq 2$ 是微不足道的，而 $Aut(\Delta _{g, n}) \cong S_n$ 当 $n \geq 1 $ 和 $(g, n) \neq (0, 4), (1, 2)$。空间 $\Delta _{g, n}$ 是一个对称的 $\Delta$-复形在 Chan、Galatius 和 Payne 的意义上，并与 Deligne-Mumford-Knudsen 模栈 $\overline {\mathcal {M}}_{g, n}$稳定的曲线。在 Massarenti 的工作之后 [J. 伦敦。数学。社会。89 (2014), pp. 131–150]，他证明了 $Aut(\overline {\mathcal {M}}_g)$ 对于 $g \geq 2$ 是微不足道的，而 $Aut(\overline {\mathcal { M}}_{g, n}) \cong S_n$ 当 $n \geq 1$ 和 $2g - 2 + n \geq 3$ 时，我们的结果意味着热带模空间 $\Delta _{g, n} $ 忠实地反映了一般 $g$ 和 $n$ 的代数模空间的对称性。

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