# Automorphisms of tropical Hassett spaces

### Sam Freedman

Brown University, Providence, USA### Joseph Hlavinka

Brown University, Providence, USA### Siddarth Kannan

Brown University, Providence, USA

## Abstract

Given an integer $g \geq 0$ and a weight vector $w \in \mathbb{Q}^n \cap (0, 1]^n$ satisfying $2g - 2 +\sum w_i > 0$, let $\Delta_{g, w}$ denote the moduli space of $n$-marked, $w$-stable tropical curves of genus $g$ and volume one. We calculate the automorphism group $\operatorname{Aut}(\Delta_{g, w})$ for $g \geq 1$ and arbitrary $w$, and we calculate the group $\operatorname{Aut}(\Delta_{0, w})$ when $w$ is heavy/light. In both of these cases, we show that $\operatorname{Aut}(\Delta_{g, w}) \cong \operatorname{Aut}(K_w)$, where $K_w$ is the abstract simplicial complex on $\{1, \ldots, n\}$ whose faces are subsets with $w$-weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space $\Delta_{g, w}$ may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space $\overline{\mathcal{M}}_{g, w}$. Following the work of Massarenti and Mella (2017) on the biregular automorphism group $\operatorname{Aut}(\overline{\mathcal{M}}_{g, w})$, we show that $\operatorname{Aut}(\Delta_{g, w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

## Cite this article

Sam Freedman, Joseph Hlavinka, Siddarth Kannan, Automorphisms of tropical Hassett spaces. Port. Math. 79 (2022), no. 1/2, pp. 163–197

DOI 10.4171/PM/2075