Symmetries of tropical moduli spaces of curves.

Put Δ_{g, n} ⊂ M_{g, n}^trop for the moduli space of stable n-marked tropical curves of genus g and volume one. We compute the automorphism group Aut(Δ_{g, n}) for all g, n ≥ 0 such that 3g ≶ 3 + n > 0. In particular, we show that Aut(Δ_g) is trivial for g ≥ 2, while Aut(Δ_{g, n}) ≅ S_n when n ≥ 1 and (g, n) ≠ (0, 4), (1, 2). The space Δ_{g, n} is a symmetric Δ-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 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(M_g) is trivial for g ≥ 2 while Aut( M_{g, n}) ≅ S_n when n ≥ 1 and 2g ≶ 2 + n ≥ 3, our result implies that the tropical moduli space Δ_{g, n} faithfully reflects the symmetries of the algebraic moduli space for general g and n. [ABSTRACT FROM AUTHOR]