Automorphisms of tropical Hassett spaces

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 $\mathrm{Aut}(\Delta_{g, w})$ for $g \geq 1$ and arbitrary $w$, and we calculate the group $\mathrm{Aut}(\Delta_{0, w})$ when $w$ is heavy/light. In both of these cases, we show that $\mathrm{Aut}(\Delta_{g, w}) \cong \mathrm{Aut}(K_w)$, 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 on the biregular automorphism group $\mathrm{Aut}(\overline{\mathcal{M}}_{g, w})$, we show that $\mathrm{Aut}(\Delta_{g, w})$ is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.


Introduction
Fix integers g, n ≥ 0 such that 2g − 2 + n > 0, let M g,n denote the moduli stack of smooth n-marked algebraic curves of genus g, and let M g,n denote its Deligne-Mumford-Knudsen compactification by stable curves.Brendan Hassett [Has03] has given a large family of alternate modular compactifications of M g,n : given a weight vector w ∈ Q n ∩ (0, 1] n satisfying 2g − 2 + n i=1 w i > 0, Hassett constructs a smooth and proper Deligne-Mumford moduli stack M g,w , birational to M g,n , which contains M g,n as a dense open substack.The points of M g,w represent n-pointed nodal curves (C, p 1 , . . ., p n ), satisfying (i) that the Q-divisor K C + w i p i is ample along each component of C, where K C is the canonical divisor of C, and (ii) if In particular, when w = (1 (n) ) is the all 1's vector, we have an equality M g,w = M g,n .
An important feature of the compactifaction M g,n ⊂ M g,n is that the boundary divisor ∂M g,n := M g,n M g,n is normal crossings.In [CGP18], Chan, Galatius, and Payne, following work of Harper [Har17] and Abramovich-Caporaso-Payne [ACP15], show how to construct the dual complex ∆(X , D) of a normal crossings divisor D on a Deligne-Mumford stack X .They study ∆(X , D) in the case where X = M g,n and D = ∂M g,n , showing that ∆(X , D) = ∆ g,n is identified with the link of the cone point in the moduli space M trop g,n of stable n-marked tropical curves of genus g.On the other hand, the complement of M g,n in Hassett's compactification M g,w is not in general normal crossings.However, if we put M g,w for the locus of smooth, but not necessarily distinctly marked, curves in M g,w , then the complement ∂M g,w := M g,w M g,w has normal crossings, and the resulting dual intersection complex ∆ g,w is the link of the cone point in the moduli space M trop g,w of n-marked, w-stable tropical curves of genus g, as has been established by Ulirsch [Uli15].
In this paper, we are interested in the automorphism groups of the complexes ∆ g,w , taken in the category of symmetric ∆-complexes, as defined in [CGP18] and recalled in Section 2. Given a weight vector w, we can form an abstract simplicial complex K w with vertex set {1, . . ., n} by declaring that a subset S ⊆ {1, . . ., n} belongs to K w if and only if i∈S w i ≤ 1; this construction was considered by Alexeev and Guy [AG08] in their work on moduli of weighted stable maps.See Figure 1 for some examples of the complex K w .Our first main theorem determines Aut(∆ g,w ) in terms of K w for g ≥ 1.
Here Aut(K w ) is viewed as a subgroup of S n := Perm({1, . . ., n}).Theorem 1.1 will be proven in Section 3, and the failure of the g = 0 case will be further explored and partially remedied in Section 4. Following Cavalieri, Hampe, Markwig, and Ranganathan [CHMR14], we refer to weight vectors satisfying the hypotheses of the following theorem as heavy/light, with m light markings and n heavy markings.
Figure 1.Examples of the simplicial complex K w .
It is also interesting to characterize the groups Aut(K w ), as in the following theorem.Since it is independent from the rest of the paper, its proof is found in Appendix A.
Theorem 1.3.Let G be a group.Then there exists n ≥ 1 and w ∈ Q n ∩ (0, 1] n such that Aut(K w ) ∼ = G if and only if G is isomorphic to the direct product of finitely many symmetric groups.
1.1.Comparison with the algebraic moduli space.For g, n ≥ 0 such that 2g − 2 + n ≥ 3, we have isomorphisms Aut(M g,n ) ∼ = Aut(∆ g,n ) ∼ = S n , following the results of [Kan] and [Mas14]; here S n acts by relabelling the marked points.The analogous result cannot be true for general weight vectors.Indeed, [Has03,Corollary 4.7] states that if w i ≤ w i for all i and the complexes K w , K w coincide outside of their 1-skeletons, then there is an isomorphism of coarse moduli spaces M g,w ∼ = M g,w .Moreover, by [MM17,Theorem 3.20], the automorphism groups of the stacks and coarse spaces agree.This implies, for example, that when w = (1 (n) , 1/2 (m) ), we have Aut(M g,w ) ∼ = S n+m .On the other hand, Theorem 1.1 states that Aut(∆ g,w ) ∼ = Aut(K w ) ∼ = S n × S m .
In [MM17], Massarenti and Mella prove that for g, n ≥ 1 such that 2g + 2 + n ≥ 3, the automorphism group of the moduli stack M g,w is given by the subgroup of S n generated by admissible transpositions.These are transpositions (i, j) such that, for all S ⊆ {1, . . ., n} with |S| ≥ 2, we have w i + w(S) ≤ 1 ⇐⇒ w j + w(S) ≤ 1, where for a subset S ⊆ {1, . . ., n} we define w(S) := i∈S w i .
The group generated by admissible transpositions acts on M g,w by relabelling the marked points, and contracting rational components which become unstable if necessary.We now show that Aut(K w ) is the subgroup of Aut(M g,w ) which preserves the locus ∂M g,w of singular curves.
Proof.Suppose first that σ is in the subgroup of S n generated by admissible transpositions but σ / ∈ Aut(K w ).Then there exists some S ⊆ {1, . . ., n} with |S| = 2, w(S) > 1, but w(σ(S)) ≤ 1; say S = {i, j}.Consider a pointed nodal curve (C, p 1 , . . ., p n ) of arithmetic genus g with two irreducible components T 1 , T 2 , so that T 2 is isomorphic to P 1 and supports the marked points p i , p j , while the other marked points are distributed distinctly on T 1 .Then σ • (C, p 1 , . . ., p n ) is obtained from (C, p 1 , . . ., p n ) by first permuting the marked points according to σ, and then contracting the component T 2 to a point so that p σ(i) = p σ(j) (this is necessary because w σ(i) + w σ(j) ≤ 1).In particular σ • (C, p 1 , . . ., p n ) is no longer a singular curve.This shows that Aut(M g,w , ∂M g,w ) is a subgroup of Aut(K w ); to finish, we simply note that when applying σ ∈ Aut(K w ) to a nodal curve (C, p 1 , . . ., p n ), there is never a need to contract any components, so Aut(K w ) preserves the boundary.
The simplicial complexes K w correspond to the chambers of the fine chamber decomposition of [Has03, Section 5] (see also [AG08, Section 2]).
In general, if D is a normal crossings divisor on a variety or DM stack X , one has a homomorphism where ∆(X , D) is the dual complex of D in X .Given Lemma 1.4, the upshot of Theorem 1.1 is that this map is an isomorphism if we specialize to D = ∂M g,w and X = M g,w : Corollary 1.5.Suppose g, n ≥ 1 with 2g − 2 + n ≥ 3, and w ∈ Q n ∩ (0, 1] n .Then the map is an isomorphism. We can also give a sufficient condition for the groups Aut(M g,w ) and Aut(K w ) to coincide.Recall that a facet of a simplicial complex is a face that is maximal with respect to inclusion.
The condition of Corollary 1.6 is sufficient but not necessary: indeed, if w = (1/2 (n) ), then Aut(M g,w ) ∼ = Aut(K w ) ∼ = S n , but K w is the complete graph on n vertices, so all of its facets are 1-dimensional.1.2.Tropical Hassett spaces excluded by Theorem 1.1.When g ≥ 1, the space ∆ g,w is nonempty as long as 3g − 3 + n > 0, so the positive genus cases not covered by Theorem 1.1 are (g, n) = (1, 1), (1, 2).When n = 1 we have ∆ 1,w = ∆ 1,1 for any w, and this space is a single point, so the automorphism group is trivial.When n = 2, so w = (w 1 , w 2 ), we have ∆ 1,w ∼ = ∆ 1,2 if w 1 + w 2 > 1, so in this case the automorphism group is trivial by [Kan,Example 2.19].When w 1 + w 2 ≤ 1, ∆ 1,w will be shown to be trivial in Example 2.10.1.3.Related work.In the special case w = (1 (n) ), the automorphism group of ∆ g,w is known to be S n : this is due to Abreu and Pacini [AP18] when g = 0, and to the third author [Kan] in arbitrary genus.Indeed, one of the main technical theorems in [Kan] is also the driving force behind the calculation in the current paper.
The topology of ∆ g,w was studied for g ≤ 1 by Cerbu et al. in [CMP + 20].When g = 0 and the weight vector w has at least two entries equal to 1, the space ∆ 0,w has the homotopy type of a wedge of spheres, possibly of varying dimension.Closed formulas for the number of spheres are known when w is heavy/light.In higher genus, the topology of ∆ g,w has been partially explored by Li, Serpente, Yun, and the third author in [KLSY20].When g ≥ 1, and for any value of w, the space ∆ g,w is shown to be simply-connected.Formulas for the Euler characteristic of ∆ g,w in terms of the combinatorics of the complex K w have also been derived.
The cone complexes M trop 0,w were studied in the context of tropical compactification in [CHMR14].The authors showed that the complex M trop 0,w can be embedded as a balanced fan Σ 0,w in a real vector space if and only if w is heavy/light.In the heavy/light case, they show that the locus M 0,w embeds into the toric variety X(Σ 0,w ), in such a way that taking the closure of the image gives Hassett's original compactification.This procedure gives an isomorphism of Chow rings A * (M 0,w ) ∼ = A * (X(Σ 0,w )), allowing for the computation of A * (M 0,w ) carried out in [KKL21].
1.4.Acknowledgements.JH was supported by a BrownConnect Collaborative SPRINT Award.SK was supported by an NSF Graduate Research Fellowship.

Graphs and ∆ g,w
We first recall the category Γ g,n of weighted stable graphs of genus g; see [CGP19, §2.1] or [Kan, §2.1] for a precise definition.An object of Γ g,n is a triple G = (G, h, m) where G is a finite connected graph, while h : V (G) → Z ≥0 and m : {1, . . ., n} → V (G) are functions; these three data are required to satisfy | + 1 denotes the first Betti number of G, and val(v) denotes the valence of the vertex v, which is the number of half-edges emanating from v. A morphism of weighted stable graphs of genus g is a composition of isomorphisms and edge-contractions.Given a morphism ϕ : G → G in Γ g,n , each edge in G has a unique preimage in G.We write ϕ * : E(G ) → E(G) for the induced map of sets.
We write Γ g,w for the full subcategory of Γ g,n whose objects are those which are w-stable.
We remark that when w = (1 (n) ), we have Γ g,w = Γ g,n .As in [Kan], it is useful to define an auxiliary groupoid Γ EL g,w , whose objects are edge-labelled w-stable graphs of genus g.

An isomorphism of pairs
commutes.
An open problem in graph theory is to classify those graphs which are determined, up to isomorphism, by their deck of edge-contractions.The reader may consult the thesis of Antoine Poirier [Poi18] for a thorough overview of this problem.The main technical tool of this paper is a solution to an easier version of this problem for the categories Γ EL g,w .Given (G, τ : we set e i = τ −1 (i) ∈ E(G), and put τ i : E(G) → [p] for the unique edge-labelling making the diagram commute, where c i : G → G/e i is the contraction of edge e i and δ i : [p − 1] → [p] is the unique order-preserving injection whose image does not contain i.
Definition 2.3.Let (G, τ ) ∈ Ob(Γ EL g,w ).We define the nonloop contraction deck of (G, τ ) to be the set of pairs Proof.In the case w = (1 (n) ), this is Theorem 4.2 in [Kan].The case of general w follows from this one, as Γ EL g,w may be identified with a full subcategory of Γ EL g,n .
2.1.Description of ∆ g,w as a functor.We will calculate Aut(∆ g,w ) in the category of symmetric ∆-complexes, as introduced by Chan, Galatius, and Payne [CGP18].Put I for the category whose objects are the sets [p] for each p ≥ 0, and whose morphisms are all injections.
Definition 2.5.A symmetric ∆-complex is a functor X : I op → Set.
A morphism of symmetric ∆-complexes is a natural transformation of functors.A symmetric ∆-complex X : I op → Set should be thought of as a set of combinatorial gluing instructions for a topological space |X|.There is a geometric realization functor given by X → |X|; see [CGP18], [Kan], or [KLSY20] for a description of this functor.
The symmetric ∆-complex description of ∆ g,w is as follows: for each p ≥ 0, we let , where π 0 denotes the set of isomorphism classes.We put [G, τ ] for the equivalence class of a Γ EL g,w -object (G, τ ), and will hereafter shorten ∆ g,w ([p]) to ∆ g,w [p].Given an injection ι : [p] → [q], we define ι * = ∆ g,w (ι) : ∆ g,w [q] → ∆ g,w [p] as follows: if [G, τ ] ∈ ∆ g,w [q], then ι * [G, τ ] is the edge-labelled graph obtained by contracting all edges in G which are not labelled by the image of ι, and then taking the induced labelling of the remaining edges which preserves their τ -ordering.
2.2.Automorphisms of ∆ g,w and the filtration by number of vertices.An automorphism of ∆ g,w is a natural isomorphism ∆ g,w → ∆ g,w .To unpack this, we will identify a generating set for the morphisms in the category I.For p ≥ 0, put so S p+1 is the group of permutations of the set {0, . . ., p}.Given α ∈ S p+1 , we write α * = ∆ g,w (α).Next, for each i ∈ [p + 1], we put δ i : [p] → [p + 1] for the unique order-preserving injection whose image does not contain the element i.We put d i := ∆ g,w (δ i ).It is apparent that any morphism ι : [p] → [q] in the category I can be factored as a sequence of maps of the form δ i , followed by some element of S q+1 .
An automorphism of ∆ g,w can therefore be understood as the data of bijections Φp commute for all α ∈ S p+1 and i ∈ [p + 1].We shall suppress the subscript and write Φ Notation 2.8.Suppose (G, τ ) ∈ Ob(Γ EL g,w ), and that we have Then, for any α ∈ S p+1 , we must have by the commutativity of (2.6).So, the action of Φ on one edge-labelling of G determines the action on all edge-labellings.We use the notation (ΦG, Φτ ) := (G , τ ); the graph ΦG is determined up to isomorphism in Γ g,w .
Remark 2.9.The group S n acts on Γ g,n : if we are given ; in this way the edges and vertices of G and σ • G are identified, so that whenever the marking i ∈ {1, . . ., n} is supported on vertex v in G, the marking σ(i) is supported on v in σ • G.A given permutation σ preserves the subcategory Γ g,w if and only if σ ∈ Aut(K w ).This gives the action of Aut(K w ) on ∆ g,w by automorphisms: Example 2.10.When w = (w 1 , w 2 ) with w 1 + w 2 ≤ 1, there are only two stable graphs in Γ 1,w with a positive number of edges: a single loop, where the single vertex supports both markings, and a pair of parallel edges, where each vertex supports one marking.Both of these graphs have a unique edge-labelling up to their automorphism groups, so we have while ∆ 1,w [p] = ∅ for p > 1, so the automorphism group is trivial in this case.The geometric realization is given by the quotient of a 1-simplex by its automorphism group S 2 .
Following [Kan], we analyze the action of Aut(∆ g,w ) by showing that it preserves the subspace V i g,w of ∆ g,w parameterizing tropical curves with at most i vertices.Each V i g,w is a subcomplex of ∆ g,w , and we have The proof that Aut(∆ g,w ) preserves this filtration is very similar to the proof of [Kan, Proposition 3.4], with some minor differences.Due to this similarity, we record the result here and relegate its proof to Appendix B.
Theorem 2.11.Let Φ ∈ Aut(∆ g,w ).Then Φ preserves the subcomplexes V i g,w for all i ≥ 1.The next theorem follows from Theorem 2.4.We omit the proof, as it is exactly the same as in the special case of w = (1 (n) ), which is [Kan, Theorem 1.5].
Theorem 2.12.Fix g ≥ 0 and a weight vector , so Theorem 1.1 specializes to the main result of [Kan].Thus we hereafter assume n ≥ 2 (and when g = 1, n ≥ 3).To prove Theorem 1.1, we first show that any Φ ∈ Aut(∆ g,w ) preserves the S n -orbit of a given simplex in V 2 g,w .
3.1.Aut(∆ g,w ) preserves S n -orbits in V 2 g,w .We want to show that for any . The first step is to show that Aut(∆ g,w ) preserves the isomorphism class of the edge-labelled graph underlying such a facet, if we forget the marking function.This motivates the following definition.
), we say that (G, τ ) and (G , τ ) are weakly isomorphic if there exists an isomorphism of weighted graphs The proof of the following lemma is the same as that of [Kan, Proposition 5.3], and is thus omitted.
Lemma 3.2.Suppose Φ ∈ Aut(∆ g,w ), and let Then, for any representatives (G, τ ), (ΦG, Φτ ), there exists a weak isomorphism ϕ : (G, τ ) (ΦG, Φτ ).Now we work towards the proof that S n -orbits of simplices in V 2 g,w [g] are preserved.For this we will need the following lemma.We adopt the convention that a 1-cycle of a graph is a loop, and a 2-cycle is a pair of parallel edges.
is the unique map such that δ i • δ i = id).Thus the claim follows by induction.(b) For i, j to label a pair of loops on the same vertex of G, we must have (i, j) By the commutativity of (2.6), we must also have (i, j) ∈ Stab S p+1 [ΦG, Φτ ], and by the first part of the lemma, i and j must both label loops of ΦG.So ΦG must have an automorphism which exchanges the loops i and j, but which fixes every other edge.If |V (ΦG)| ≥ 3, this is only possible if i and j label loops on the same vertex of ΦG.When |V (ΦG)| = 2, the claim follows from Lemma 3.2, and the claim is clear when |V (ΦG)| = 1.
Lemmas 3.2 and 3.3 give us a criterion for checking whether Aut(∆ g,w ) preserves the weak isomorphism class of a given simplex, and at the level of V 2 g,w , Lemma 3.2 means that Aut(∆ g,w ) acts on an edge-labelled stable graph by at most changing the marking function.We would like to show that this redistribution preserves the number of markings on each vertex.It is useful to introduce notation parameterizing the facets of V 2 g,w .Fix a vertex set {v 1 , v 2 }.For two integers k, such that k, ≥ 0 and k + ≤ g, we fix B k, to be a graph with vertex set {v 1 , v 2 }, where v 1 and v 2 are connected by g − (k + ) + 1 edges, so that v 1 supports k loops while v 2 supports loops.
By construction, B k, has genus g and g + 1 edges, and we have graph isomorphisms B k, ∼ = B ,k .Up to isomorphism, any facet of the subcomplex V 2 g,w is a marked, edge labeled version of B k, for some k, .Given a subset A ⊆ {1, . . ., n}, we put B k, A for an n-marked version of B k, , where a vertex with k loops supports the elements of A and the other vertex supports the elements of A c .We will use the boldface notation B k, A when B k, A defines a Γ g,w -object.Fixing choices of edge-labellings π k, : Throughout the remainder of this section, we will tacitly change the choice of π k, for a given k, if it is necessary.To prove that Aut(∆ g,w ) ∼ = Aut(K w ), it suffices to show that for any Φ ∈ Aut(∆ g,w ), there exists a unique element σ ∈ Aut(K w ) such that for all k, , A.

Recalling that the B k,
A are precisely the facets of V 2 g,w we can see, using that Φ( ), that Theorem 3.4 extends to the preservation of S n -orbits in V 2 g,w as a whole.
3.2.Finishing the proof of Theorem 1.1.First we deal with the case n = 2.In this case, we have Aut(K w ) ∼ = S 2 , and g ≥ 2. Given x ∈ {1, . . ., n} we have that the graph {x} is always stable.Given Φ ∈ Aut(∆ g,w ), there exists a unique element σ ∈ S 2 such that Proof.This proof follows from that of [Kan, Proposition 5.13(b)]: in that argument, no reference is made to w-unstable simplices in order to show that a given w-stable simplex is fixed.
Proof.We will show that for S ⊆ {1, . . ., n}, we have that w(S) > 1 if and only if w(σ(S)) > 1.By switching σ with σ −1 , it suffices to just prove one direction.We may suppose that some S with w(S) > 1 exists, because otherwise Aut(K w ) ∼ = S n .Since w(S) > 1, the graph B g,0 S c is stable.Moreover, the graph B g,0 S c shares an expansion with B 0,0 x if and only if x ∈ S c .Therefore, if we let Φ(S c ) be as determined by Theorem 3.4, we have Φ(S c ) = σ(S c ).This implies that the graph B g,0 σ(S c ) is stable, i.e. that w((σ(S c )) c ) = w(σ(S)) > 1, as we wanted to show.Given the result of Lemma 3.6, Theorem 1.1 is rendered equivalent to the following result.x ] is fixed, we must have that either then by stability B 3 = {x} and B 1 = ∅.Upon contracting edge j in both graphs, we would then have that the S n -orbit of [B 0,0 A ] is not preserved by Φ, which is impossible.Therefore, we have B 1 ∪ B 3 = {x} c and B 2 = {x}.Moreover, since S n -orbits of graphs with two vertices are preserved, we have Since n ≥ 3, this case only arises when n ≥ 4 and n is even.We treat the n = 4 case and the n ≥ 5 case separately.If n ≥ 5, then the set of graphs B 0,0 S such that |S| = 2 sharing an expansion with B 0,0 A are precisely those S such that S ⊆ A or S ⊆ A c .Since n ≥ 5, we know all of the B 0,0 S with |S| = 2 are fixed, hence for any choice of Φ This implies that A = Φ(A) or A = Φ(A) c , so in particular we have that [B 0,0 A ] is fixed by Φ. Finally consider the case when n = 4 and |A| = 2. Say |A| = {x, y} and A c = {u, v}, and suppose for sake of contradiction we have Φ[B 0,0 {x,y} ] = [B 0,0 {x,v} ].Then consider an expansion T 2 of B 0,0 {x,y} as in Figure 3, with an edge labelling τ so that . ., n} denote the marking sets on the vertices of Φ[T 2 ] as in Figure 3. x ] = [B 0,0 x ], we must have which is a contradiction.Therefore ] by hypothesis, it follows that B 2 = {v} and B 3 = {u, y}.From this we may conclude that This clearly contradicts our hypothesis if g = 1, so we may now suppose that g ≥ 2.
Finally, consider the simplex [T 3 ] and its image under Φ in Figure 4, where the edge-labelling is such that d . ., n} be the marking sets of Φ[T 3 ] as in Figure 4. Proof.If k = 0, then [B k,0 A ] is fixed by Proposition 3.9.When k = g, B g,0 A shares a common expansion with B 0,0 x if and only if x ∈ A, so [B g,0 A ] is fixed because all of the simplices [B 0,0 x ] are fixed.So suppose that 1 ≤ k < g and that A is nonempty.
For a pair [B k,0 A ] and [B 0,0 x ] with x ∈ A consider the graph G k,A of Figure 5, where the {x, A−{x}} multiedge has cardinality k + 1 and contains an edge labelled j, and the {A c , x} multiedge has cardinality g − k and contains an edge labelled h.Give such a graph an edge labeling τ such that . ., n} be the marking sets on the vertices of G k,A as in Figure 5.
As such, assume for contradiction that x ∈ C 1 .Then Proof.If either of k, = 0 then [B k, A ] is fixed by Proposition 3.9.Similarly, by Theorem 3.4, we may always assume A, A c nonempty.So, assume first that k, ≥ 1, k + < g.For every pair [B k, A ] and [B 0, x ] with x ∈ A consider the graph G k, ,A of Figure 6, where the {x, A − {x}} multiedge has cardinality k + 1, and the {A c , x} multiedge has cardinality g − k − .Give such a graph an edge labeling τ such that By hypothesis, and k are nonzero, so neither of the vertices adjacent ] for some choice of edge labeling, a contradiction to Lemma 3.2.This means that the loops in Φ[G k, ,A ] are adjacent to edges h and i.Thus, by Lemma 3.3 applied to all remaining edges of Φ[G k, ,A ], we have that Φ[G k, ,A ] is weakly isomorphic to [G k, ,A ], so the situation is as depicted in Figure 6; let D 1 , D 2 , D 3 ⊆ {1, . . ., n} be the marking sets on the vertices as in the figure .By Proposition 3.9, Give such a graph an edge-labelling τ such that A ]: this is immediate ignoring the loops, while the loops cannot be adjacent to edge j lest A ] for some edge labeling of B g,0 A .Let E 1 , E 2 , E 3 ⊆ {1, . . ., n} be the markings on the vertices of Φ[G k, ,A ] as in Figure 7. ,A ] and its image under Φ Then, one can see that x ] by Proposition 3.9.As such, since E 1 ∪ E 2 = Φ(A), we have that x ∈ A implies x ∈ Φ(A), and the result follows.
The preceding three propositions combine to prove Theorem 3.7.
Proof of Theorem 3.7.Suppose given Φ ∈ Aut(∆ g,w ) such that Φ fixes all the simplices [B 0,0 x ].Proposition 3.8 shows then that Φ fixes all [B 0,0 A ], then Proposition 3.9 shows that Φ fixes the [B k,0 A ], and then Proposition 3.10 shows that Φ fixes all the [B k, A ]. Finally, we conclude this section by indicating how Theorem 1.1 follows from its results.
Proof of Theorem 1.1.By Theorem 2.12, it suffices to show that for any Φ ∈ Aut(∆ g,w ), there exists a unique element σ ∈ Aut(K w ) such that Φ = σ when restricted to V 2 g,w .Given such Φ, there exists a unique permutation σ ∈ S n such that Φ = σ on the n simplices [B 0,0 x ], by Theorem 3.4.Lemma 3.6 then implies that σ ∈ Aut(K w ).Then Theorem 3.7 gives that Φ • σ −1 acts as the identity on the facets of V 2 g,w , from which it follows that Φ • σ −1 fixes the whole subcomplex V 2 g,w .Thus Φ = σ on V 2 g,w and the proof is complete.

The genus 0 case
When g = 0, Theorem 1.1 fails for general w.We will first give some counterexamples, and then proceed to show that the theorem still holds when g = 0 and w is assumed to be heavy/light.4.1.Counterexamples to Theorem 1.1 when g = 0. We now give an infinite family of counterexamples to Theorem 1.1 in the case g = 0.
Proposition 4.1.For each integer k ≥ 1, set where Proof.The space ∆ 0,w k consists of N (k) disjoint vertices; this is because the only w k -stable trees consist of only one edge, where each vertex supports k +1 markings.This proves that Aut(∆ 0,w k ) ∼ = S N (k) .Clearly Aut(K w k ) ∼ = S 2k+2 , so it only remains to prove that N (k) > 2k + 2 for all k ≥ 2. We do this by induction.When k = 2, we have N (2) = 10 and 2k + 2 = 6.Now suppose it is known that 2k k > 4k.
Remark 4.4.In [CHMR14], the tropical moduli space M trop 0,w is realized, for heavy/light w = ( (m) , 1 (n) ), as the Bergman fan B(G w ) of the graphic matroid of the reduced weight graph G w associated to w.The vertex set of G w is {1, 2, . . ., m + n − 1}, and edge (i, j) is included whenever w i + w j > 1.The space ∆ 0,w can be constructed as the link of M trop 0,w at its cone point, so in particular we have Aut(∆ 0,w ) ∼ = Aut(B(G w )).The fan B(G w ) carries actions of the groups Aut(G w ) and Aut(I(G w )), where I(G w ) denotes the independence complex of the graph G w .In general, we have Aut(G w ) ∼ = S m × S n−1 , while a general description of Aut(I(G w )) eludes the authors.In the case n = 2, the graph G w is a star with m leaves, and I(G w ) is a standard (m − 1)simplex, so we have that Aut(G w ) ∼ = Aut(I(G w )) ∼ = S m .By Theorem 4.3, in this case both groups are strictly smaller than the automorphism group Aut(B(G w )) ∼ = S m × S 2 .4.3.∆ 0,w as a flag complex.When g = 0 and w ∈ Q n ∩ (0, 1] n with w i > 2, the objects in Γ 0,w are automorphism-free, and hence ∆ 0,w may be realized as a simplicial complex.Given some A ⊆ {1, . . ., n} with w(A), w(A c ) > 1, we put B A for a Γ 0,w -object with one edge, such that one vertex supports the elements of A, and the other supports the elements of A c .A collection of vertices {B A 1 , . . ., B A k } spans a (k − 1)-simplex of ∆ 0,w if and only if there exists a Γ 0,w -object G with precisely k edges e 1 , . . ., e k , such that G/{e i } c ∼ = B A i for all i; here G/{e} c indicates the graph obtained from G by contracting all edges except for e.
We claim that ∆ 0,w is a flag complex.This claim when w = (1 (n) ) is due to N. Giansiracusa [Gia16], and its proof is based on the Buneman Splits-Equivalence Theorem [SS03, Theorem 3.1.4],which we state here in a form compatible with our notation: The analogous theorem for ∆ 0,w follows from the following observation: a graph G in Γ 0,n lies in Γ 0,w if and only if G/{e} c lies in Γ 0,w for all e ∈ E(G).Thus, if we are given a collection {B A 1 , . . ., B A k } of vertices of ∆ 0,w such that each pair of them spans a 1-simplex of ∆ 0,w , then we can use Theorem 4.5 to guarantee that there exists some graph G in Γ 0,n such that G has precisely k edges e 1 , . . ., e k , and such that G/{e i } c ∼ = B A i for all i.By our observation, we actually have G ∈ Ob(Γ 0,w ), hence {B A 1 , . . ., B A k } spans a simplex of ∆ 0,w .As such, we have the following corollary of Theorem 4.5.
To prove Theorem 4.7, we will show that any automorphism of ∆ (1) 0,w can be completely described by its action on graphs of the form B i,j := B {i,j} , where i ∈ {1, . . .m} and j ∈ {m + 1, . . ., m + n}.Graphs of this form will be called special.Special graphs have the maximal number of expansions among all graphs in ∆ (1) 0,w : Lemma 4.8.Consider the same hypotheses as Theorem 4.7.For a graph G, let exp(G) denote the number of isomorphism classes of expansions of G with precisely one more edge than G. Then for all graphs B i,j as above and for all vertices with equality if and only if B A = B i ,j for possibly different indices i ∈ {1, . . .m} and j ∈ {m + 1, . . ., m + n}.
The proof of Lemma 4.8 amounts to a somewhat tedious application of basic calculus, and can be found in Appendix C. Establishing an analogue of this lemma for arbitrary weight vectors seems to be the principal obstruction to determining the groups Aut(∆ 0,w ) in general.
Proof of Theorem 4.7.The desired isomorphism is given by the map where Φ (σ,τ ) is the automorphism of ∆ (1) 0,w that relabels light points using the permutation σ ∈ S m and the heavy points with the permutation τ ∈ S n .We must show that F is both injective and surjective: F is injective.Supposing that Φ (σ,τ ) acts as the identity on ∆ (1) 0,w , we must show that (σ, τ ) is the identity permutation.We use that Φ (σ,τ ) in particular fixes each special graph B i,j .As we are assuming m + n ≥ 5, the graph B i,j has at least 3 marked points on its other endpoint.It follows that {σ(i), τ (j)} = {i, j}, or σ(i) = i and τ (j) = j.This demonstrates that (σ, τ ) is the identity permutation.
For any such graph B A , we can decompose A into light and heavy weights as A = A L A H , where A L ⊂ {1, . . ., m} and A H ⊂ {m+1, . . ., m+n}.Similarly we can decompose A C into disjoint sets A C L and A C H .Note that the set of special graphs incident to B A is incident is then precisely {B i,j } ∪ {B i ,j }, where (i, j) ∈ A L × A H and (i , j ) ∈ A C L × A C H .In general if B A is incident to special vertices {B i,j }, then A can be recovered up to complement from the pairs (i, j).Indeed, start with any such neighbor B i,j ; without loss of generality, i and j are supported on the left-hand endpoint of B A .We can read off the rest of the markings on this vertex as follows.The left-hand light indices i are those for which B i ,j is incident to B A .Similarly, the left-hand heavy indices j are those for which B i ,j is incident to B A for all left-hand light indices i .All of the weights {i } ∪ {j } either make up A or A C , so we conclude that B A is uniquely determined by its special neighbors.
In summary, we know that Φ (σ,τ ) • Ψ −1 fixes all special neighbors of B A , and that B A is the unique one-edge graph incident to all of these special neighbors.It follows that Φ (σ,τ ) • Ψ −1 fixes B A as well, so Φ (σ,τ ) = Ψ and F is surjective.
Appendix A. Proof of Theorem 1.3 In this appendix we prove Theorem 1.3, restated here.
Theorem.Let G be a group.Then there exists n ≥ 1 and w ∈ Q n ∩ (0, 1] n such that Aut(K w ) ∼ = G if and only if G is isomorphic to the direct product of finitely many symmetric groups.
We will first prove that Aut(K w ) is always isomorphic to a product of symmetric groups, i.e. that it is generated by transpositions.We require a preliminary lemma.
Lemma A.1.Suppose H is a subgroup of S n , and that for all σ ∈ H and i ∈ {1, . . ., n}, we have (i, σ(i)) ∈ H. Then H is generated by transpositions.
Proof.We want to show that given σ ∈ H, we can write σ = τ 1 • • • τ k where each τ i ∈ H is a transposition.First consider the case where σ = (i 1 , . . ., i r ) is a cycle.Then we have Each transposition (i 1 , σ j (i 1 )) lies in H, so the above gives a decomposition of the desired form for σ.To remove the assumption that σ is a cycle, we decompose into disjoint cycles and run the same argument.
Proof.By the preceding lemma, it suffices to prove that if σ ∈ Aut(K w ) satisfies σ(i) = j, then τ = (i, j) ∈ Aut(K w ).Indeed, suppose toward a contradiction that τ / ∈ Aut(K w ).Then there exists S ⊆ {1, . . ., n} such that S ∈ K w but τ (S) / ∈ K w , i.e. w(S) ≤ 1, but w(τ (S)) > 1.If i, j ∈ S, or i, j ∈ S c , then w(S) = w(τ (S)) so it must be that exactly one of i, j lies in S, suppose WLOG that i ∈ S and j / ∈ S. Write S = { 1 , . . ., p , i} = L ∪ {i} where L = { 1 , . . ., p }.For any natural number k ≥ 0, we have σ k ∈ Aut(K w ), so we must have but using that L = τ (L) and j = σ(i), we have so in particular w σ k+1 (i) > w σ k (i) for all k ≥ 0. This is a contradiction as σ has finite order.We conclude that τ ∈ Aut(K w ), as we wanted to show.
The following lemma allows us to symmetrize the weight data with respect to the action of Aut(K w ).
Lemma A.3.Suppose n ≥ 2 and w ∈ Q n ∩ (0, 1] n .Then there exists some ŵ ∈ Q n ∩ (0, 1] n such that Proof.Since Aut(K w ) is generated by transpositions, it suffices to show that if τ = (i, j) ∈ Aut(K w ), then the weight vector ŵ obtained from w by changing both w i and w j to (w i + w j )/2 satisfies K ŵ = K w .Indeed, suppose w(S) ≤ 1 to show that ŵ(S) ≤ 1.If both i, j are contained in S or S c , then w(S) = ŵ(S), so it suffices to consider the case where i ∈ S and j / ∈ S; write S = L ∪ {i} where i, j / ∈ L. Then ŵ(S) = w(L) + w i + w j 2 ≤ w(L) + max(w i , w j ) ≤ 1, since τ ∈ Aut(K w ).This shows that any S ∈ K w satisfies S ∈ K ŵ.Conversely suppose ŵ(S) ≤ 1 to show that w(S) ≤ 1. Again we may focus on the case where i ∈ S but j / ∈ S; write S = L ∪ {i}, where i, j / ∈ L. Suppose for contradiction that Then we must also have w(τ (S)) = ŵ(L) + w j > 1.
It follows that ŵ(S) = ŵ(L) + which is a contradiction.Thus S ∈ K w , and we are done.
Proposition A.2 gives one direction of Theorem 1.3: since Aut(K w ) is generated by transpositions, it is always isomorphic to a direct product of symmetric groups, and this product has to be finite as Aut(K w ) is finite.We have left to show that an arbitrary finite direct product of symmetric groups can be realized in this way.
Proof of Theorem 1.3.Suppose S n i for some integers n i ≥ 1.We prove that there exists w such that Aut(K w ) ∼ = G by induction on k.When k = 1, we simply take w to be an all 1's vector.For the inductive step, suppose we have some vector ŵ such that Aut(K ŵ) ∼ = k−1 i=1 S n i , in order to construct w such that Aut(K w ) ∼ = G.We may assume that n k > 1.
For an arbitrary vector w, we say an index i ∈ {1, . . ., n} is heavy in w if w i + w j > 1 for all indices j = i.We say an index i is light in w if for all S ⊆ {1, . . ., n} with w(S) < 1, we have w(S) + w i ≤ 1.If i is heavy, respectively light, in w, then we have that (i, j) ∈ Aut(K w ) if and only if j is also heavy, respectively light, in w.Moreover, by the previous lemma, there exists some ε > 0 such that if w is obtained from w by changing all heavy weights to 1 and light weights to ε, then K w = K w .
If ŵ does not contain any heavy, respectively light, weights, then we can construct w by adding n k heavy, respectively light, weights to ŵ, in which case we have Aut(K w ) ∼ = G.Otherwise, if ŵ contains both heavy and light weights, we can assume all of the heavy weights are equal to 1 and the light weights are equal to ε for some ε > 0. Also by the previous lemma we may assume that whenever σ ∈ Aut(K ŵ) satisfies σ(i) = j, we have ŵi = ŵj .Then we set • G has no bridges where a bridge of a graph G is a non-loop edge which is not contained in any cycles (see Figure 3 in [Kan] for an example of such a graph in general).Then the necessary Γ g,w object G can be constructed by choosing n points on the interiors of edges of G, and putting a vertex supporting a marking at each chosen point.The graph G cannot contract to a graph which has a bridge, so the only graph with one edge that it contracts to is R 1 .
To prove that the simplex [R g ] is preserved, we first preserve that bridge indices are preserved, as in the following lemma.
That is, an edge e is a bridge of G if and only if upon contracting all edges in G besides e, we do not get a loop.The lemma now follows from this description of B G τ and Proposition B.1.We can now prove that automorphisms preserve the simplex [R g ].
Proof.Suppose G is a maximal graph of Γ g,w , with the property that every bridge of G is either a loop or a bridge (it is straightforward to construct examples of such G for all g ≥ 1 and weight vectors w).
, then B indexes the bridges in both G and G .Since bridges are contained in all spanning trees, the edges indexed by B in G must be contained in some spanning tree of G .On the other hand, we know the edges indexed by B in G form a spanning tree of G. Since G and G have the same number of vertices, they have the same number of edges in a spanning tree.Therefore the edges indexed by B in G form a spanning tree.Whenever we contract a spanning tree in a Γ g,w -object of first Betti number g, the resulting graph is R g .In particular, we have and the proof is complete.We restate the lemma for convenience: Lemma.Consider the same hypotheses as Theorem 4.7.For a graph G, let exp(G) denote the number of isomorphism classes of expansions of G with precisely one more edge than G. Then for all graphs B i,j as above and for all vertices B A ∈ ∆ (1) 0,w , exp(B i,j ) ≥ exp(B A ), with equality if and only if B A = B i ,j for possibly different indices i ∈ {1, . . .m} and j ∈ {m + 1, . . ., m + n}.

Proof.
In what follows let B A be a graph with one edge, where we think of the weights in A as occupying the left-hand vertex.Set x := |A|, and suppose that there are y light weights in A.
We are interested in maximizing the number of expansions of B A .Left expansions are bijective correspondence with subsets S of A such that w(S) > 1 and |A S| > 0. There are 2 x −2 y −(x−y)−1 such subsets S: 2 x total subsets of A, minus the 2 y subsets consisting solely of light weights (including the empty subset), minus the x − y singleton subsets consisting solely of one heavy weight, minus the subset A itself.Repeating the counting argument on the other side, we conclude that  These inequalities arise as follows.First, the stability condition requires that both vertices of B A have at least two weights on them, so x = |A| is bounded by 2 and m + n − 2. Second, the number of light weights on either vertex cannot exceed m, the total number of light weights.Finally, there must be at least one heavy weight on either vertex, so that the number of left-hand heavy weights x − y is at least 1 and at most n − 1.
To prevent some of the above ranges from collapsing, it is convenient to address the case n = 2 separately from n ≥ 3: n = 2.As there are exactly two heavy weights, each vertex of B A supports one of them.In other words, we have y = x − 1.We are now maximizing the function f (x, x − 1) = 2 x − 2 x−1 + 2 (m+2)−x − 2 m−(x−1) = 2 x−1 + 2 m−x+1 .over the interval 2 ≤ x ≤ m.As the second derivative d 2 dx 2 (2 x−1 + 2 m−x+1 ) = 2 −x−1 log 2 (2)(2 m+2 + 4 x ) is non-negative on [2, m], f (x, x − 1) is convex and thus achieves its maximum value on its endpoint x = 2 as desired.(The other endpoint x = m corresponds to the complement B A C = B A .) n ≥ 3. First, we look for critical points on the interior of the region.We compute the gradient as ∇f = log(2)2 x − 2 m+n−x , 2 −y log(2)(2 m − 4 y ) .Setting the partial derivatives equal to 0, we find that there is one critical point located at ((m + n)/2, m/2).
We now optimize f over the boundary.Note that there is a symmetry originating from exchanging the two vertices of B A ; symbolically, this is the involution (x, y) → (m + n − x, m − y).Therefore it suffices to optimize f over only half of the boundary, i.e.only over the left-hand equalities.
We first look for critical points on the interiors of these segments, and second consider the values of f at their endpoints.
• First, note that f (2, y) = 2 m+n−2 − 2 m−y − 2 y + 4 has a critical point at y = m/2.As m ≥ 2, this critical point is outside of the interior of the interval for y. • Second, note that f (2, y) = f (x, 0) = 2 m+n−x − 2 m + 2 x − 1 + 4 has one critical point at x = (m + n)/2.This point is interior when 2 < (m + n)/2 and (m + n)/2 < n − 1.That is, there is a critical point interior to this edge whenever n > m + 2. • Finally, note that f (1 + y, y) = 2 m+n−y−1 − 2 m−y + 2 y has a critical point when 2 m + 2 2y = 2 m+n−1 .Since m = m + n − 1 and 2y = m + n − 1 immediately leads to a contradiction, when 2y = m all of the exponents in this equation are distinct.Thus there is no integer solution by the uniqueness of binary representations.In case 2y = m the equation reduces to m + 1 = m − n − 1, or n = 2.As we are assuming n ≥ 3, this edge contains no critical points.

Figure 2 .
Figure 2. The simplex [T 1 ] and its image under Φ

Figure 3 .
Figure 3.The simplex [T 2 ] and its image under Φ

Figure 4 .
Figure 4.The simplex [T 3 ] and its image under Φ Then, as we have Φ[B 0,g−1 y

Figure 5 .
Figure 5.The simplex [G k,A ] and its image under Φ , and thus D 3 = A {x}.This implies that Φ[G k, ,A ] = [G k, ,A ], and so each contraction of [G k, ,A ]) is fixed under Φ as well.Now say k + = g, with k, > 0. Recall that by Theorem 3.4, we may always assume that |A|, |A c | ≥ 1.Then, for every pair [B k, A ] and [B 0, x ] with x ∈ A, consider the graph G k, ,A of Figure 7, where the {x, A {x}} multiedge has cardinality k + 1.
and a proper subset of indices S ⊂ [p], we put d S [G, τ ] for the face of [G, τ ] obtained by contracting all edges labelled by elements of S. From the commutativity of diagram 2.7, it can be shown that for any automorphism Φ of ∆ g,n , we have Φd S [G, τ ] = d S Φ[G, τ ].With this notation in place, we can characterize B G τ as follows: Let τ : E(G) → [p] be any edge-labelling of G, and put [G , τ ] = Φ[G, τ ].Then we claim G also has the property that all of its bridges are either loops or bridges.Indeed, G must also be maximal, so b 1 (G ) = b 1 (G) = g, and hence we have |V (G )| = |V (G)|.By Lemma B.2, G has the same number of bridges as G, and if we set B As per the discussion preceding Proposition B.1, Theorem 2.11 is a corollary of Proposition B.3.Appendix C. Proof of Lemma 4.8 exp(B A ) = 2 x − 2 y − (x − y) − 1 + 2 (m+n)−x − 2 m−y − [(m + n − x) − (m − y)] − 1 = 2 x − 2 y + 2 m+n−x − 2 m−y − 2 − n expansions total.It therefore suffices to maximizef (x, y) := 2 x − 2 y + 2 (m+n)−x − 2 m−yover a domain that includes all permissible integer values of (x, y).Such a domain is determined by the three inequalities 2 ≤ x ≤ (m + n) − 2, 0 ≤ y ≤ m, and 1 ≤ (x − y) ≤ n − 1; see Figure10.

Figure 10 .
Figure 10.The domain of f (x, y) and only if it indexes a k-cycle of ΦG via Φτ ; (b) a subset {i, j} ⊆ [p] indexes a pair of loops on the same vertex of G if and only if it indexes a pair of loops on the same vertex of ΦG.Proof.We prove each part separately.(a) When k = 1, the claim is true as an index i labels a loop of G if and only if G/e i has the same number of vertices as G, and Φd i [G, τ ] = d i [ΦG, Φτ ].Since Φ preserves the number of vertices, it follows that i must label a loop index of ΦG.Now S labels a k-cycle of G if and only if, for all i ∈ S, the set δ