Automorphisms of tropical Hassett spaces
AAUTOMORPHISMS OF TROPICAL HASSETT SPACES
SAM FREEDMAN, JOSEPH HLAVINKA, AND SIDDARTH KANNAN
Abstract.
Given an integer g ≥ w ∈ Q n ∩ (0 , n satisfying 2 g − (cid:80) w i > g,w denote the moduli space of n -marked, w -stable tropical curves of genus g and volume one.We calculate the automorphism group Aut(∆ g,w ) for g ≥ w , and we calculate thegroup Aut(∆ ,w ) when w is heavy/light . In both of these cases, we show that Aut(∆ g,w ) ∼ = Aut( K w ),where K w is the abstract simplicial complex on { , . . . , n } whose faces are subsets with w -weightat most 1. We show that these groups are precisely the finite direct products of symmetric groups.The space ∆ g,w may also be identified with the dual complex of the divisor of singular curves inthe algebraic Hassett space M g,w . Following the work of Massarenti and Mella [MM17] on thebiregular automorphism group Aut( M g,w ), we show that Aut(∆ g,w ) is naturally identified withthe subgroup of automorphisms which preserve the divisor of singular curves. Contents
1. Introduction 12. Graphs and ∆ g,w
53. Calculation of Aut(∆ g,w ) for g ≥ Introduction
Fix integers g, n ≥ g − 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 compact-ification by stable curves. Brendan Hassett [Has03] has given a large family of alternate modularcompactifications of M g,n : given a weight vector w ∈ Q n ∩ (0 , n satisfying2 g − n (cid:88) i =1 w i > , 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 nodalcurves ( C, p , . . . , p n ), satisfying (i) that the Q -divisor K C + (cid:80) w i p i is ample along each componentof C , where K C is the canonical divisor of C , and (ii) if p i = · · · = p i r , then w i + · · · + w i r ≤ 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 (cid:114) 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 a r X i v : . [ m a t h . C O ] F e b SAM FREEDMAN, JOSEPH HLAVINKA, AND SIDDARTH KANNAN 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 conepoint 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 generalnormal crossings. However, if we put M g,w for the locus of smooth, but not necessarily distinctlymarked, curves in M g,w , then the complement ∂ M g,w := M g,w (cid:114) M g,w has normal crossings, and the resulting dual intersection complex ∆ g,w is the link of the cone pointin the moduli space M trop g,w of n -marked, w -stable tropical curves of genus g , as has been establishedby Ulirsch [Uli15].In this paper, we are interested in the automorphism groups of the complexes ∆ g,w , taken inthe category of symmetric ∆ -complexes , as defined in [CGP18] and recalled in Section 2. Givena weight vector w , we can form an abstract simplicial complex K w with vertex set { , . . . , n } bydeclaring that a subset S ⊆ { , . . . , n } belongs to K w if and only if (cid:80) i ∈ S w i ≤
1; this constructionwas considered by Alexeev and Guy [AG08] in their work on moduli of weighted stable maps. SeeFigure 1 for some examples of the complex K w . Our first main theorem determines Aut(∆ g,w ) interms of K w for g ≥ Theorem 1.1.
Let g ≥ and suppose w ∈ Q n ∩ (0 , n for some n such that g − n ≥ . Then Aut(∆ g,w ) ∼ = Aut( K w ) , where Aut( K w ) acts by permuting the markings. Here Aut( K w ) is viewed as a subgroup of S n := Perm( { , . . . , n } ). Theorem 1.1 will be provenin Section 3, and the failure of the g = 0 case will be further explored and partially remedied inSection 4. Following Cavalieri, Hampe, Markwig, and Ranganathan [CHMR14], we refer to weightvectors satisfying the hypotheses of the following theorem as heavy/light , with m light markingsand n heavy markings. Theorem 1.2.
Suppose n, m ≥ , with n + m ≥ , and put w = ( ε ( m ) , ( n ) ) where ε ≤ /m . Thenwe have Aut(∆ ,w ) ∼ = Aut( K w ) ∼ = S m × S n . Heavy/light Hassett spaces are of particular interest: they are also studied in [BM13], [Cha16],[KKL21], [LM04], and [MOP11]. , , , (cid:0) , , , (cid:1) (cid:0) , , , (cid:1) (cid:0) , , , (cid:1) S S S S × S wK w Aut( K w ) 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 isindependent from the rest of the paper, its proof is found in Appendix A. UTOMORPHISMS OF TROPICAL HASSETT SPACES 3
Theorem 1.3.
Let G be a group. Then there exists n ≥ and w ∈ Q n ∩ (0 , n such that Aut( K w ) ∼ = G if and only if G is isomorphic to the direct product of finitely many symmetric groups. Comparison with the algebraic moduli space.
For g, n ≥ g − n ≥
3, wehave 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. Theanalogous result cannot be true for general weight vectors. Indeed, [Has03, Corollary 4.7] statesthat if w i ≤ w (cid:48) i for all i and the complexes K w , K w (cid:48) coincide outside of their 1-skeletons, then thereis an isomorphism of coarse moduli spaces M g,w ∼ = M g,w (cid:48) . Moreover, by [MM17, Theorem 3.20],the automorphism groups of the stacks and coarse spaces agree. This implies, for example, thatwhen w = (1 ( n ) , / ( m ) ), we have Aut( M g,w ) ∼ = S n + m . On the other hand, Theorem 1.1 states thatAut(∆ g,w ) ∼ = Aut( K w ) ∼ = S n × S m .In [MM17], Massarenti and Mella prove that for g, n ≥ g + 2 + n ≥
3, the auto-morphism group of the moduli stack M g,w is given by the subgroup of S n generated by admissibletranspositions . These are transpositions ( i, j ) such that, for all S ⊆ { , . . . , n } with | S | ≥
2, wehave w i + w ( S ) ≤ ⇐⇒ w j + w ( S ) ≤ , where for a subset S ⊆ { , . . . , n } we define w ( S ) := (cid:88) 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 thatAut( K w ) is the subgroup of Aut( M g,w ) which preserves the locus ∂ M g,w of singular curves. Lemma 1.4.
Suppose g, n ≥ with g − n ≥ , and fix w ∈ Q n ∩ (0 , n . Then Aut( M g,w , ∂ M g,w ) ∼ = Aut( K w ) , where Aut( K w ) acts by permuting the markings.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 ⊆ { , . . . , n } with | S | = 2, w ( S ) >
1, but w ( σ ( S )) ≤
1; say S = { i, j } . Consider a pointed nodal curve ( C, p , . . . , p n ) of arithmetic genus g with two irreduciblecomponents T , T , so that T is isomorphic to P and supports the marked points p i , p j , while theother marked points are distributed distinctly on T . Then σ · ( C, p , . . . , p n ) is obtained from( C, p , . . . , p n ) by first permuting the marked points according to σ , and then contracting thecomponent T to a point so that p σ ( i ) = p σ ( j ) (this is necessary because w σ ( i ) + w σ ( j ) ≤ σ · ( C, p , . . . , p n ) is no longer a singular curve. This shows that Aut( M g,w , ∂ M g,w ) isa subgroup of Aut( K w ); to finish, we simply note that when applying σ ∈ Aut( K w ) to a nodalcurve ( C, p , . . . , p n ), there is never a need to contract any components, so Aut( K w ) preserves theboundary. (cid:3) 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 homomorphismAut( X , D ) → Aut(∆( X , D )) , where ∆( X , D ) is the dual complex of D in X . Given Lemma 1.4, the upshot of Theorem 1.1 isthat this map is an isomorphism if we specialize to D = ∂ M g,w and X = M g,w : SAM FREEDMAN, JOSEPH HLAVINKA, AND SIDDARTH KANNAN
Corollary 1.5.
Suppose g, n ≥ with g − n ≥ , and w ∈ Q n ∩ (0 , n . Then the map Aut( M g,w , ∂ M g,w ) → Aut(∆ g,w ) 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. Corollary 1.6.
Suppose g, n ≥ with g − n ≥ , and w ∈ Q n ∩ (0 , n . Then, if K w has no -dimensional facets, then Aut( M g,w ) = Aut( K w ) . Proof.
It suffices to show that every admissible transposition τ = ( i, j ) is in Aut( K w ). Suppose S ⊆ { , . . . , n } satisfies w ( S ) ≤
1, in order to show that w ( τ ( S )) ≤
1. If both i, j ∈ S or i, j ∈ S c ,then w ( τ ( S )) = w ( S ), so we suppose without loss of generality that i ∈ S while j ∈ S c . Then,if | S | ≥
3, by the definition of admissible transposition, we must have w ( τ ( S )) ≤
1. If | S | = 2,then S forms a 1-simplex of K w , and cannot be a facet. Thus there exists some T ⊆ { , . . . , n } such that S (cid:40) T and w ( T ) ≤
1. Since | T | ≥
3, we have w ( τ ( T )) ≤
1, and w ( τ ( S )) < w ( τ ( T )), so w ( τ ( S )) ≤ τ ∈ Aut( K w ), finishing the proof. (cid:3) The condition of Corollary 1.6 is sufficient but not necessary: indeed, if w = (1 / ( n ) ), thenAut( M g,w ) ∼ = Aut( K w ) ∼ = S n , but K w is the complete graph on n vertices, so all of its facets are1-dimensional.1.2. Tropical Hassett spaces excluded by Theorem 1.1.
When g ≥
1, the space ∆ g,w isnonempty as long as 3 g − n >
0, so the positive genus cases not covered by Theorem 1.1 are( g, n ) = (1 , , (1 , n = 1 we have ∆ ,w = ∆ , for any w , and this space is a singlepoint, so the automorphism group is trivial. When n = 2, so w = ( w , w ), we have ∆ ,w ∼ = ∆ , if w + w >
1, so in this case the automorphism group is trivial by [Kan, Example 2.19]. When w + w ≤
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 knownto 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 forcebehind the calculation in the current paper.The topology of ∆ g,w was studied for g ≤ et al. in [CMP + g = 0 andthe weight vector w has at least two entries equal to 1, the space ∆ ,w has the homotopy type ofa wedge of spheres, possibly of varying dimension. Closed formulas for the number of spheres areknown when w is heavy/light. In higher genus, the topology of ∆ g,w has been partially exploredby Li, Serpente, Yun, and the third author in [KLSY20]. When g ≥
1, and for any value of w , thespace ∆ g,w is shown to be simply-connected. Formulas for the Euler characteristic of ∆ g,w in termsof the combinatorics of the complex K w have also been derived.The cone complexes M trop0 ,w were studied in the context of tropical compactification in [CHMR14].The authors showed that the complex M trop0 ,w can be embedded as a balanced fan Σ ,w in a realvector space if and only if w is heavy/light. In the heavy/light case, they show that the locus M ,w embeds into the toric variety X (Σ ,w ), in such a way that taking the closure of the imagegives Hassett’s original compactification. This procedure gives an isomorphism of Chow rings A ∗ ( M ,w ) ∼ = A ∗ ( X (Σ ,w )), allowing for the computation of A ∗ ( M ,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.
UTOMORPHISMS OF TROPICAL HASSETT SPACES 5 Graphs and ∆ g,w We first recall the category Γ g,n of weighted stable graphs of genus g ; see [CGP19, § § g,n is a triple G = ( G, h, m ) where G is a finiteconnected graph, while h : V ( G ) → Z ≥ and m : { , . . . , n } → V ( G ) are functions; these three dataare required to satisfy b ( G ) + (cid:88) v ∈ V ( G ) h ( v ) = g, and 2 h ( v ) − v ) + | m − ( v ) | > v ∈ V ( G ). In the above, b ( G ) = | E ( G ) | − | V ( G ) | + 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 emanatingfrom v . A morphism of weighted stable graphs of genus g is a composition of isomorphisms andedge-contractions. Given a morphism ϕ : G → G (cid:48) in Γ g,n , each edge in G (cid:48) has a unique preimagein G . We write ϕ ∗ : E ( G (cid:48) ) → E ( G ) for the induced map of sets. Definition 2.1.
Given w ∈ Q n ∩ (0 , n , say G ∈ Ob(Γ g,n ) is w -stable if for all v ∈ V ( G ), we have2 h ( v ) − v ) + w ( m − ( v )) > . 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 anauxiliary groupoid Γ EL g,w , whose objects are edge-labelled w -stable graphs of genus g . Definition 2.2.
Define Γ EL g,w to be the groupoid of pairs ( G , τ ) where G ∈ Ob(Γ g,w ) and τ : E ( G ) → [ p ] is a bijection, where for an integer p ≥
0, we define[ p ] = { , . . . , p } . An isomorphism of pairs ϕ : ( G , τ ) → ( G (cid:48) , τ (cid:48) ) is an isomorphism G → G (cid:48) such that the diagram E ( G (cid:48) ) E ( G )[ p ] ϕ ∗ τ (cid:48) τ commutes.An open problem in graph theory is to classify those graphs which are determined, up to iso-morphism, by their deck of edge-contractions. The reader may consult the thesis of AntoinePoirier [Poi18] for a thorough overview of this problem. The main technical tool of this paperis a solution to an easier version of this problem for the categories Γ EL g,w . Given ( G , τ : E ( G ) → [ p ]) ∈ Ob(Γ EL g,w ) and i ∈ [ p ], we set e i = τ − ( i ) ∈ E ( G ), and put τ i : E ( G ) → [ p ] for the uniqueedge-labelling making the diagram E ( G /e i ) E ( G )[ p −
1] [ p ] c ∗ i τ i τδ i commute, where c i : G → G /e i is the contraction of edge e i and δ i : [ p − → [ p ] is the uniqueorder-preserving injection whose image does not contain i . SAM FREEDMAN, JOSEPH HLAVINKA, AND SIDDARTH KANNAN
Definition 2.3.
Let ( G , τ ) ∈ Ob(Γ EL g,w ). We define the nonloop contraction deck of ( G , τ ) tobe the set of pairs D G τ := { (( G /e i , τ i ) , i ) | e i is not a loop of G } ⊆ Ob(Γ EL g,w ) × [ p ] . Given two lists D = { (( G i , τ i ) , i ) | i ∈ J } , D = { (( H i , π i ) , i ) | i ∈ J } of Γ EL g,w objects indexedby J , J ⊆ [ p ], we write D ∼ = D if J = J , and ( G i , τ i ) ∼ = ( H i , π i ) for all i ∈ J . Theorem 2.4.
Suppose ( G , τ ) , ( G (cid:48) , τ (cid:48) ) ∈ Ob(Γ EL g,w ) with b ( G ) = b ( G (cid:48) ) = g . Suppose further that D G τ ∼ = D G (cid:48) τ (cid:48) and that | V ( G ) | = | V ( G (cid:48) ) | ≥ . Then ( G , τ ) ∼ = ( G (cid:48) , τ (cid:48) ) .Proof. In the case w = (1 ( n ) ), this is Theorem 4.2 in [Kan]. The case of general w follows from thisone, as Γ EL g,w may be identified with a full subcategory of Γ EL g,n . (cid:3) 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 whoseobjects 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 instructionsfor a topological space | X | . There is a geometric realization functor given by X (cid:55)→ | 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∆ g,w ([ p ]) = { ( G , τ ) ∈ π (Γ EL g,w ) | | E ( G ) | = p + 1 } , where π denotes the set of isomorphism classes. We put [ G , τ ] for the equivalence class of aΓ 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 theedge-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 automorphismof ∆ g,w is a natural isomorphism ∆ g,w → ∆ g,w . To unpack this, we will identify a generating setfor the morphisms in the category I. For p ≥
0, put S p +1 := Hom I ([ p ] , [ p ]) , so S p +1 is the group of permutations of the set { , . . . , 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 whoseimage 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 bysome element of S q +1 .An automorphism of ∆ g,w can therefore be understood as the data of bijectionsΦ = { Φ p : ∆ g,w [ p ] → ∆ g,w [ p ] } p ≥ , such that the diagrams(2.6) ∆ g,w [ p ] ∆ g,w [ p ]∆ g,w [ p ] ∆ g,w [ p ] Φ p α ∗ α ∗ Φ p UTOMORPHISMS OF TROPICAL HASSETT SPACES 7 and(2.7) ∆ g,w [ p + 1] ∆ g,w [ p + 1]∆ g,w [ p ] ∆ g,w [ p ] Φ p +1 d i d i Φ p commute for all α ∈ S p +1 and i ∈ [ p + 1]. We shall suppress the subscript and write Φ[ G , τ ] forΦ p [ G , τ ]. Notation 2.8.
Suppose ( G , τ ) ∈ Ob(Γ EL g,w ), and that we haveΦ[ G , τ ] = [ G (cid:48) , τ (cid:48) ] . Then, for any α ∈ S p +1 , we must haveΦ[ G , τ ◦ α ] = [ G (cid:48) , τ (cid:48) ◦ α ]by the commutativity of (2.6). So, the action of Φ on one edge-labelling of G determines the actionon all edge-labellings. We use the notation (Φ G , Φ τ ) := ( G (cid:48) , τ (cid:48) ); the graph Φ G is determined upto isomorphism in Γ g,w . Remark 2.9.
The group S n acts on Γ g,n : if we are given G = ( G, h, m ) ∈ Ob(Γ g,n ), we put σ · G = ( G, h, m ◦ σ − ); in this way the edges and vertices of G and σ · G are identified, so thatwhenever the marking i ∈ { , . . . , n } is supported on vertex v in G , the marking σ ( i ) is supportedon 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: σ · [ G , τ ] = [ σ · G , τ ]. Example 2.10.
When w = ( w , w ) with w + w ≤
1, there are only two stable graphs in Γ ,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 aunique edge-labelling up to their automorphism groups, so we have | ∆ ,w [0] | = | ∆ ,w [1] | = 1while ∆ ,w [ p ] = ∅ for p >
1, so the automorphism group is trivial in this case. The geometricrealization is given by the quotient of a 1-simplex by its automorphism group S .Following [Kan], we analyze the action of Aut(∆ g,w ) by showing that it preserves the subspace V ig,w of ∆ g,w parameterizing tropical curves with at most i vertices. Each V ig,w is a subcomplex of∆ g,w , and we have V g,w ⊆ V g,w ⊆ · · · ⊆ V g − ng,w = ∆ g,w . The proof that Aut(∆ g,w ) preserves this filtration is very similar to the proof of [Kan, Proposition3.4], with some minor differences. Due to this similarity, we record the result here and relegate itsproof to Appendix B.
Theorem 2.11.
Let Φ ∈ Aut(∆ g,w ) . Then Φ preserves the subcomplexes V ig,w for all i ≥ . The next theorem follows from Theorem 2.4. We omit the proof, as it is exactly the same as inthe special case of w = (1 ( n ) ), which is [Kan, Theorem 1.5]. Theorem 2.12.
Fix g ≥ and a weight vector w ∈ Q n ∩ (0 , n , where g − (cid:80) w i > . Thenthe restriction map Aut(∆ g,w ) → Aut( V g,w ) is an injection. SAM FREEDMAN, JOSEPH HLAVINKA, AND SIDDARTH KANNAN Calculation of
Aut(∆ g,w ) for g ≥ g, n ≥ w ∈ Q n ∩ (0 , n , where 2 g − n ≥
3. If n = 1, then ∆ g,w ∼ = ∆ g, , soTheorem 1.1 specializes to the main result of [Kan]. Thus we hereafter assume n ≥ g = 1, n ≥ ∈ Aut(∆ g,w ) preserves the S n -orbitof a given simplex in V g,w .3.1. Aut(∆ g,w ) preserves S n -orbits in V g,w . We want to show that for any [ G , τ ] in V g,w , theaction of Φ ∈ Aut(∆ g,n ) preserves the S n -orbit of [ G , τ ]. It suffices to show this in the case where[ G , τ ] is a facet of V g,w , i.e. [ G , τ ] ∈ V g,w [ g ]. The first step is to show that Aut(∆ g,w ) preservesthe isomorphism class of the edge-labelled graph underlying such a facet, if we forget the markingfunction. This motivates the following definition. Definition 3.1.
Given two objects ( G , τ ) , ( G (cid:48) , τ (cid:48) ) of Γ EL g,w with G = ( G, h, m ) and G (cid:48) = ( G (cid:48) , h (cid:48) , m (cid:48) ),we say that ( G , τ ) and ( G (cid:48) , τ (cid:48) ) are weakly isomorphic if there exists an isomorphism of weightedgraphs ϕ : ( G, h ) → ( G (cid:48) , h (cid:48) )making the diagram E ( G (cid:48) ) E ( G )[ p ] τ (cid:48) ϕ ∗ τ commute. Such a map ϕ is called a weak isomorphism of pairs , and is denoted with a dashedarrow ϕ : ( G , τ ) (cid:57)(cid:57)(cid:75) ( G (cid:48) , τ (cid:48) ) . The proof of the following lemma is the same as that of [Kan, Proposition 5.3], and is thusomitted.
Lemma 3.2.
Suppose Φ ∈ Aut(∆ g,w ) , and let [ G , τ ] ∈ V g,w [ g ] . Then, for any representatives ( G , τ ) , (Φ G , Φ τ ) , there exists a weak isomorphism ϕ : ( G , τ ) (cid:57)(cid:57)(cid:75) (Φ G , Φ τ ) . Now we work towards the proof that S n -orbits of simplices in V g,w [ g ] are preserved. For this wewill need the following lemma. We adopt the convention that a 1-cycle of a graph is a loop, and a2-cycle is a pair of parallel edges. Lemma 3.3.
Let Φ ∈ Aut(∆ g,w ) , and suppose [ G , τ ] ∈ ∆ g,w [ p ] . Then:(a) a subset S ⊆ [ p ] indexes a k -cycle of G via τ if and only if it indexes a k -cycle of Φ G via Φ τ ;(b) a subset { i, j } ⊆ [ p ] indexes a pair of loops on the same vertex of G if and only if it indexes apair of loops on the same vertex of Φ G .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 thesame number of vertices as G , and Φ d i [ G , τ ] = d i [Φ G , Φ τ ]. Since Φ preserves the number ofvertices, it follows that i must label a loop index of Φ G . Now S labels a k -cycle of G if andonly if, for all i ∈ S , the set δ i ( S ) labels a ( k − d i [ G , τ ] (here, δ i : [ p ] (cid:114) { i } → [ p − 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 ) ∈ Stab S p +1 [ G , τ ].By the commutativity of (2.6), we must also have ( i, j ) ∈ Stab S p +1 [Φ G , Φ τ ], and by the firstpart of the lemma, i and j must both label loops of Φ G . So Φ G must have an automorphismwhich exchanges the loops i and j , but which fixes every other edge. If | V (Φ G ) | ≥
3, this is
UTOMORPHISMS OF TROPICAL HASSETT SPACES 9 only possible if i and j label loops on the same vertex of Φ G . When | V (Φ G ) | = 2, the claimfollows from Lemma 3.2, and the claim is clear when | V (Φ G ) | = 1. (cid:3) Lemmas 3.2 and 3.3 give us a criterion for checking whether Aut(∆ g,w ) preserves the weakisomorphism class of a given simplex, and at the level of V 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 liketo show that this redistribution preserves the number of markings on each vertex. It is useful tointroduce notation parameterizing the facets of V g,w .Fix a vertex set { v , v } . For two integers k, (cid:96) such that k, (cid:96) ≥ k + (cid:96) ≤ g , we fix B k,(cid:96) to bea graph with vertex set { v , v } , where v and v are connected by g − ( k + (cid:96) ) + 1 edges, so that v supports k loops while v supports (cid:96) loops.By construction, B k,(cid:96) has genus g and g + 1 edges, and we have graph isomorphisms B k,(cid:96) ∼ = B (cid:96),k .Up to isomorphism, any facet of the subcomplex V g,w is a marked, edge labeled version of B k,(cid:96) for some k , (cid:96) . Given a subset A ⊆ { , . . . , n } , we put B k,(cid:96)A for an n -marked version of B k,(cid:96) , wherea 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,(cid:96)A when B k,(cid:96)A defines a Γ g,w -object. Fixing choices ofedge-labellings π k,(cid:96) : E ( B k,(cid:96) ) → [ g ], we put [ B k,(cid:96)A ] = [ B k,(cid:96)A , π k,(cid:96) ] for the resulting simplex in V g,w [ g ].Throughout the remainder of this section, we will tacitly change the choice of π k,(cid:96) for a given k, (cid:96) ifit 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 thatΦ[ B k,(cid:96)A ] = [ σ · B k,(cid:96)A ] = [ B k,(cid:96)σ ( A ) ]for all k, (cid:96), A .Following [Kan], we now observe that for a fixed choice of A , there always exists some (notnecessarily unique) permutation σ A ∈ S n such that Φ[ B k,(cid:96)A ] = [ B k,(cid:96)σ A ( A ) ]: Theorem 3.4.
Suppose g, n ≥ with g − n ≥ and w ∈ Q n ∩ (0 , n . Then for all Φ ∈ Aut(∆ g,w ) and B k,(cid:96)A , there exists some Φ( A ) ⊆ { , . . . , n } such that | A | = | Φ( A ) | and Φ[ B k,(cid:96)A ] =[ B k,(cid:96) Φ( A ) ] . Moreover, the choice of such Φ( A ) is unique unless ( k, (cid:96) ) = (0 , and | A | = n/ .Proof. The argument is exactly the same as [Kan, Section 5.2], except in the case ( k, (cid:96) ) = (0 , A ⊆ { , . . . , n } , we let µ ( A ) = |{ G ∈ π (Γ g,w ) | | V ( G ) | = 3 , G has a 3-cycle, ≤ B , A }| . For any A ⊆ { , . . . , n } , it is straightforward to compute µ ( A ) = 2 | A | + 2 n −| A | + Constant , cf. [Kan, Proposition 5.2]. By Lemma 3.2, there exists some C ⊆ { , . . . , n } such thatΦ[ B , A ] = [ B , C ] , Lemma 3.3 implies that we must have µ ( A ) = µ ( C ). In particular, we must have either | A | = | C | or | A | = n − | C | . Then unless | A | = n/
2, there is a unique choice between Φ( A ) = C, C c suchthat Φ[ B , A ] = [ B , A ) ] and | A | = | Φ( A ) | . If | A | = n/
2, this choice is only determined up tocomplementation. (cid:3)
Recalling that the B k,(cid:96)A are precisely the facets of V g,w we can see, using that Φ( d i [ B k,(cid:96)A ]) = d i (Φ[ B k,(cid:96)A ]), that Theorem 3.4 extends to the preservation of S n -orbits in V g,w as a whole. Finishing the proof of Theorem 1.1.
First we deal with the case n = 2. In this case, wehave Aut( K w ) ∼ = S , and g ≥
2. Given x ∈ { , . . . , n } we have that the graph B g − , x := B g − , { x } is always stable. Given Φ ∈ Aut(∆ g,w ), there exists a unique element σ ∈ S such thatΦ[ B g − , x ] = [ B g − , σ ( x ) ]for x = 1 , Proposition 3.5.
Suppose n = 2 , g ≥ , and w ∈ Q ∩ (0 , . Then if Φ ∈ Aut(∆ g,w ) fixes thesimplices [ B g − , x ] for x = 1 , , then Φ | V g,w = Id | V g,w .Proof. This proof follows from that of [Kan, Proposition 5.13(b)]: in that argument, no referenceis made to w -unstable simplices in order to show that a given w -stable simplex is fixed. (cid:3) Given the result of Proposition 3.5, we assume that n ≥ g ≥
1, the graph B , x := B , { x } is stable for each x ∈ { , . . . , n } . Theorem 3.4 gives a unique permutation σ ∈ S n such thatΦ[ B , x ] = [ B , σ ( x ) ] for all x ∈ { , . . . , n } . We will first show that σ ∈ Aut( K w ), and then that Φacts as σ on all elements of V g,w [ g ]. For the remainder of the paper, by an expansion of a graph H ∈ Ob(Γ g,w ) we will mean a graph G with | E ( G ) | > | E ( H ) | admitting a Γ g,w -morphism to H . Lemma 3.6.
Suppose g ≥ , n ≥ , and w ∈ Q n ∩ (0 , n , and suppose Φ ∈ Aut(∆ g,n ) . Let σ ∈ S n be the unique permutation such that Φ[ B , x ] = [ B , σ ( x ) ] for all x ∈ { , . . . , n } . Then σ ∈ Aut( K w ) .Proof. We will show that for S ⊆ { , . . . , n } , we have that w ( S ) > w ( σ ( S )) > σ with σ − , it suffices to just prove one direction. We may suppose that some S with w ( S ) > K w ) ∼ = S n . Since w ( S ) >
1, the graph B g, S c is stable.Moreover, the graph B g, S c shares an expansion with B , x if and only if x ∈ S c . Therefore, if welet 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, σ ( S c ) is stable, i.e. that w (( σ ( S c )) c ) = w ( σ ( S )) >
1, as we wanted to show. (cid:3)
Given the result of Lemma 3.6, Theorem 1.1 is rendered equivalent to the following result.
Theorem 3.7.
Fix g ≥ , n ≥ , and say that Φ ∈ Aut(∆ g,w ) fixes each simplex [ B , x ] . Then Φ fixes all of the simplices [ B k,(cid:96)A ] . Theorem 3.7 is further broken up into the three intermediate results Proposition 3.8, Proposition3.9, and Proposition 3.10, whose statements and proofs make up the remainder of this section.
Proposition 3.8.
Fix g ≥ , n ≥ , and suppose Φ ∈ Aut(∆ g,w ) fixes the simplices [ B , x ] for all x ∈ { , . . . , n } . Then, Φ fixes the simplices [ B , A ] for all A ⊆ { , . . . , n } .Proof. If | A | or | A c | = 0, then [ B , A ] is fixed by by Theorem 3.6. If | A | or | A c | = 1 then B , A = B , x for some x ∈ { , . . . , n } , so it’s fixed by hypothesis. So, assume that | A | and | A c | ≥
2, and for nowwe also assume that | A | (cid:54) = n/
2, so in particular | A | (cid:54) = | A c | and there is a unique Φ( A ) ⊆ { , . . . , n } with | Φ( A ) | = | A | and Φ[ B , A ] = [ B , A ) ]. For each x ∈ A , consider the graph T of Figure 2, withsome edge-labelling τ such that d i [ T , τ ] = [ B , x ] and d j [ T , τ ] = [ B , A ]; we put [ T ] := [ T , τ ].Then, by Lemma 3.3, we have that Φ[ T ] is weakly isomorphic to [ T ]. Let B , B , B ⊆ { , . . . , n } be the markings on the vertices of Φ[ T ] as indicated by Figure 2. UTOMORPHISMS OF TROPICAL HASSETT SPACES 11 A c A (cid:114) { x } xi j B B B i j Φ Figure 2.
The simplex [ T ] and its image under ΦThen, since [ B , x ] is fixed, we must have that either B ∪ B = { x } or B ∪ B = { x } c . Butif B ∪ B = { x } , then by stability B = { x } and B = ∅ . Upon contracting edge j in bothgraphs, we would then have that the S n -orbit of [ B , A ] is not preserved by Φ, which is impossible.Therefore, we have B ∪ B = { x } c and B = { x } . Moreover, since S n -orbits of graphs with twovertices are preserved, we have | A c | + 1 = | B | + | B | , so | B | = | A c | , hence | B | = | A | −
1. Itfollows that B ∪ B = Φ( A ), hence x ∈ Φ( A ). It follows that A ⊆ Φ( A ), so in fact A = Φ( A ) byTheorem 3.4.It is only left to show that Φ[ B , A ] = [ B , A ] when | A | = n/
2. Since n ≥
3, this case only ariseswhen n ≥ n is even. We treat the n = 4 case and the n ≥ n ≥
5, thenthe set of graphs B , S such that | S | = 2 sharing an expansion with B , A are precisely those S suchthat S ⊆ A or S ⊆ A c . Since n ≥
5, we know all of the B , S with | S | = 2 are fixed, hence for anychoice of Φ( A ) such that Φ[ B , A ] = [ B , A ) ], we must have that (cid:18) A (cid:19) ∪ (cid:18) A c (cid:19) = (cid:18) Φ( A )2 (cid:19) ∪ (cid:18) Φ( A ) c (cid:19) . This implies that A = Φ( A ) or A = Φ( A ) c , so in particular we have that [ B , 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 supposefor sake of contradiction we have Φ[ B , { x,y } ] = [ B , { x,v } ] . Then consider an expansion T of B , { x,y } as in Figure 3, with an edge labelling τ so that d i [ T , τ ] =[ B , { x,y } ] and d j [ T , τ ] = [ B , x ]; we put [ T ] = [ T , τ ]. We let B (cid:48) , B (cid:48) , B (cid:48) ⊆ { , . . . , n } denote themarking sets on the vertices of Φ[ T ] as in Figure 3. x vyj Φ ui B (cid:48) B (cid:48) ji B (cid:48) Figure 3.
The simplex [ T ] and its image under ΦThen, since Φ[ B , x ] = [ B , x ], we must have d j Φ[ T ] = [ B , x ], so either B (cid:48) ∪ B (cid:48) = { x } or B (cid:48) = { x } . If B (cid:48) ∪ B (cid:48) = { x } , then by stability we have B (cid:48) = { x } while B (cid:48) = ∅ , but thenΦ[ B , { x,y } ] = d i Φ[ T ] = [ B , ∅ ] , which is a contradiction. Therefore B (cid:48) = { x } while B (cid:48) ∪ B (cid:48) = { x } c . Since d i Φ[ T ] = [ B , { x,v } ] byhypothesis, it follows that B (cid:48) = { v } and B (cid:48) = { u, y } . From this we may conclude thatΦ[ B ,g − y ] = [ B ,g − v ] . This clearly contradicts our hypothesis if g = 1, so we may now suppose that g ≥ T ] and its image under Φ in Figure 4, where the edge-labelling issuch that d i [ T ] = [ B , y ] and d j [ T ] = [ B ,g − y ]. Let B (cid:48)(cid:48) , B (cid:48)(cid:48) , B (cid:48)(cid:48) ⊆ { , . . . , n } be the marking setsof Φ[ T ] as in Figure 4. y v Φ i j ux B (cid:48)(cid:48) B (cid:48)(cid:48) i jB (cid:48)(cid:48) h h Figure 4.
The simplex [ T ] and its image under ΦThen, as we have Φ[ B ,g − y ] = [ B ,g − v ], it must be that B (cid:48)(cid:48) = { v } while B (cid:48)(cid:48) ∪ B (cid:48)(cid:48) = { v } c .On the other hand, since Φ[ B , y ] = [ B , y ], we must have d i Φ[ T ] = [ B , y ], hence B (cid:48)(cid:48) = { y } and B (cid:48)(cid:48) = { x, v } . From this it follows that Φ d h [ T ] = B , { x,v } . This is a contradiction as d h [ T ] = [ B , ∅ ]must be fixed by Theorem 3.4. This completes the proof. (cid:3) The next step is to show that any automorphism which fixes the simplices [ B , A ] must also fixthe simplices [ B k, A ]. Proposition 3.9.
Fix g ≥ , n ≥ , and suppose Φ ∈ Aut(∆ g,w ) fixes all simplices of the form [ B , A ] . Then, Φ fixes all simplices of the form [ B k, A ] with k ≤ g .Proof. If k = 0, then [ B k, A ] is fixed by Proposition 3.9. When k = g , B g, A shares a commonexpansion with B , x if and only if x ∈ A , so [ B g, A ] is fixed because all of the simplices [ B , x ] arefixed. So suppose that 1 ≤ k < g and that A is nonempty.For a pair [ B k, A ] and [ B , 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 hascardinality g − k and contains an edge labelled h . Give such a graph an edge labeling τ such that d i [ G k,A ] = [ B , x ] ,d j [ G k,A ] = [ B k, A ] , and d h [ G k,A ] = [ B g − k − , A c ∪{ x } ];we put [ G k,A ] := [ G k,A , τ ]. Then, by Lemma 3.3, we have that Φ[ G k,A ] is weakly isomorphic to[ G k,A ]. Let C , C , C ⊆ { , . . . , n } be the marking sets on the vertices of G k,A as in Figure 5.If we can show that x (cid:54)∈ C , we’re done: we must have C ∪ C = Φ( A ), where Φ( A ) ⊆ { , . . . , n } is uniquely determined by Theorem 3.4. So if x ∈ C ∪ C we have x ∈ Φ( A ) which gives that A ⊆ Φ( A ), and the result follows as | A | = | Φ( A ) | .As such, assume for contradiction that x ∈ C . Then d i [ G k,A ] = [ B , x ], so C = ∅ , and C = { x } c . Since n − | C ∪ C | = | Φ( A ) | = | A | , this gives | A c | = | C | = 1. Thus d h Φ[ G k,A ] = [ B g − k − , { ,...,n } ] . UTOMORPHISMS OF TROPICAL HASSETT SPACES 13 A c A (cid:114) { x } xi j Φ h C C C i jh Figure 5.
The simplex [ G k,A ] and its image under ΦHowever, we have d h [ G k,A ] = [ B g − k − , A c ∪{ x } ] , so we must have n = | A c | + 1 = 2, a contradiction to our hypothesis that n ≥ (cid:3) By symmetry, Proposition 3.9 gives us that any [ B ,(cid:96)A ] is also fixed. Proposition 3.10.
Fix g ≥ , n ≥ , and suppose Φ ∈ Aut(∆ g,w ) fixes all simplices of the form [ B k, A ] with k < g . Then, Φ fixes all simplices of the form [ B k,(cid:96)A ] .Proof. If either of k, (cid:96) = 0 then [ B k,(cid:96)A ] is fixed by Proposition 3.9. Similarly, by Theorem 3.4, wemay always assume A, A c nonempty. So, assume first that k, (cid:96) ≥ k + (cid:96) < g . For every pair [ B k,(cid:96)A ]and [ B ,(cid:96)x ] with x ∈ A consider the graph G k,(cid:96),A of Figure 6, where the { x, A − { x }} multiedgehas cardinality k + 1, and the { A c , x } multiedge has cardinality g − k − (cid:96) . Give such a graph anedge labeling τ such that d i [ G k,(cid:96),A ] = [ B ,(cid:96)x ], d j [ G k,(cid:96),A ] = [ B k,(cid:96)A ], and d h [ G k,(cid:96),A ] = [ B g − k − , A c ∪{ x } ]; weput [ G k,(cid:96),A ] := [ G k,(cid:96),A , τ ]. By hypothesis, (cid:96) and k are nonzero, so neither of the vertices adjacent A c A (cid:114) { x } xi j Φ h D D D i jh Figure 6.
The simplex [ G k,(cid:96),A ] and its image under Φto edge j in Φ( G k,(cid:96),A ] can support (cid:96) loops: if either did, then d j Φ[ G k,(cid:96),A ] = [ B k + (cid:96), A ) ] for somechoice of edge labeling, a contradiction to Lemma 3.2. This means that the (cid:96) loops in Φ[ G k,(cid:96),A ]are adjacent to edges h and i . Thus, by Lemma 3.3 applied to all remaining edges of Φ[ G k,(cid:96),A ], wehave that Φ[ G k,(cid:96),A ] is weakly isomorphic to [ G k,(cid:96),A ], so the situation is as depicted in Figure 6; let D , D , D ⊆ { , . . . , n } be the marking sets on the vertices as in the figure.By Proposition 3.9, d i Φ[ G k,(cid:96),A ] = Φ[ B ,(cid:96)x ] = [ B ,(cid:96)x ] . Further, by the same, d h Φ[ G k,(cid:96),A ] = Φ[ B g − k − , A c ∪{ x } ] = [ B g − k − , A c ∪{ x } ] . Since (cid:96) (cid:54) = 0, this implies that D = { x } and D ∪ D = A c ∪{ x } , so D = A c , and thus D = A (cid:114) { x } .This implies that Φ[ G k,(cid:96),A ] = [ G k,(cid:96),A ], and so each contraction of [ G k,(cid:96),A ]) is fixed under Φ as well. Now say k + (cid:96) = g , with k, (cid:96) >
0. Recall that by Theorem 3.4, we may always assume that | A | , | A c | ≥
1. Then, for every pair [ B k,(cid:96)A ] and [ B ,(cid:96)x ] with x ∈ A , consider the graph G (cid:48) k,(cid:96),A of Figure7, where the { x, A (cid:114) { x }} multiedge has cardinality k + 1.Give such a graph an edge-labelling τ such that d i ([ G (cid:48) k,(cid:96),A ]) = [ B ,(cid:96)x ] and d j ([ G (cid:48) k,(cid:96),A ]) = [ B k,(cid:96)A ];we put [ G (cid:48) k,(cid:96),A ] := [ G (cid:48) k,(cid:96),A , τ ]. By Lemma 3.3, Φ([ G (cid:48) k,(cid:96),A ]) is weakly isomorphic to [ G (cid:48) k,(cid:96),A ]: this isimmediate ignoring the loops, while the loops cannot be adjacent to edge j lest d j (Φ([ G (cid:48) k,(cid:96),A ])) =[ B g, A ] for some edge labeling of B g, A . Let E , E , E ⊆ { , . . . , n } be the markings on the verticesof Φ[ G (cid:48) k,(cid:96),A ] as in Figure 7. Φ j i (cid:96)x A (cid:114) { x } A c j i (cid:96)E E E Figure 7.
The simplex [ G (cid:48) k,(cid:96),A ] and its image under ΦThen, one can see that E = { x } becauseΦ d i [ G (cid:48) k,(cid:96),A ] = Φ[ B ,(cid:96)x ] = [ B ,(cid:96)x ]by Proposition 3.9. As such, since E ∪ E = Φ( A ), we have that x ∈ A implies x ∈ Φ( A ), and theresult follows. (cid:3) 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 , x ].Proposition 3.8 shows then that Φ fixes all [ B , A ], then Proposition 3.9 shows that Φ fixes the[ B k, A ], and then Proposition 3.10 shows that Φ fixes all the [ B k,(cid:96)A ]. (cid:3) 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 ), thereexists a unique element σ ∈ Aut( K w ) such that Φ = σ when restricted to V g,w . Given such Φ,there exists a unique permutation σ ∈ S n such that Φ = σ on the n simplices [ B , x ], by Theorem3.4. Lemma 3.6 then implies that σ ∈ Aut( K w ). Then Theorem 3.7 gives that Φ ◦ σ − acts as theidentity on the facets of V g,w , from which it follows that Φ ◦ σ − fixes the whole subcomplex V g,w .Thus Φ = σ on V g,w and the proof is complete. (cid:3) The genus 0 case
When g = 0, Theorem 1.1 fails for general w . We will first give some counterexamples, and thenproceed 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 coun-terexamples to Theorem 1.1 in the case g = 0. Proposition 4.1.
For each integer k ≥ , set w k = (1 /k ) (2 k +2) . Then Aut(∆ g,w k ) ∼ = S N ( k ) , where N ( k ) := 12 · (cid:18) k + 2 k + 1 (cid:19) . Moreover, we have N ( k ) > k + 2 for k ≥ and N (1) = 3 < , so Aut(∆ g,w k ) (cid:54)∼ = Aut( K w k ) for all k ≥ . UTOMORPHISMS OF TROPICAL HASSETT SPACES 15
Proof.
The space ∆ ,w k consists of N ( k ) disjoint vertices; this is because the only w k -stable treesconsist of only one edge, where each vertex supports k +1 markings. This proves that Aut(∆ ,w k ) ∼ = S N ( k ) . Clearly Aut( K w k ) ∼ = S k +2 , so it only remains to prove that N ( k ) > k + 2 for all k ≥ k = 2, we have N (2) = 10 and 2 k + 2 = 6. Now suppose it isknown that (cid:18) kk (cid:19) > k. We then have (cid:18) k + 2 k + 1 (cid:19) = (2 k + 2)(2 k + 1)( k + 1) · (cid:18) kk (cid:19) > (2 k + 2)(2 k + 1)( k + 1) · k. Observe that (2 k + 2)(2 k + 1)( k + 1) = 4 k + 6 k + 1 k + 2 k + 1 > > k , for k ≥
2. Hence (cid:18) k + 2 k + 1 (cid:19) > (cid:18) k (cid:19) k = 4 k + 4 , as desired. (cid:3) Example 4.2.
The family of examples provided by Proposition 4.1 are 0-dimensional. A 1-dimensional example where Aut(∆ ,w ) (cid:54)∼ = Aut( K w ) is given by w = (1 / (3) , / (3) ). In this case wehave Aut( K w ) ∼ = S × S , but Aut(∆ ,w ) is isomorphic to the wreath product S (cid:111) S . See Figures8 and 9. This also gives an example where Aut(∆ ,w ) is not isomorphic to a direct product ofsymmetric groups, which cannot happen when g ≥
1, by Theorem 1.1 and Theorem 1.3. B , B , , B , , B , , B , B , , B , , B , , B , B , , B , , B , , Figure 8.
The tropical moduli space ∆ ,w for w = (1 / (3) , / (3) ). Figure 9.
The simplicial complex K w for w = (1 / (3) , / (3) ). Calculation of
Aut(∆ ,w ) for heavy/light w . In this section, we will remedy the genus 0failure of Theorem 1.1 when w is heavy/light , i.e. when w = ( ε ( m ) , ( n ) ) for ε ≤ /m . Theorem 4.3.
Suppose m + n ≥ and n, m ≥ , and set w = ( ε ( m ) , ( n ) ) for any ε ≤ /m . Then Aut(∆ ,w ) ∼ = Aut( K w ) ∼ = S m × S n . To prove Theorem 4.3, we will describe ∆ ,w as a flag complex , i.e. the maximal simplicialcomplex on its 1-skeleton. This allows us to calculate Aut(∆ ,w ) by instead calculating the auto-morphism group of its 1-skeleton. Remark 4.4.
In [CHMR14], the tropical moduli space M trop0 ,w is realized, for heavy/light w =( (cid:15) ( m ) , ( 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 { , , . . . , m + n − } , and edge ( i, j ) is included whenever w i + w j >
1. The space ∆ ,w can be constructed as the link of M trop0 ,w at its cone point, soin particular we have Aut(∆ ,w ) ∼ = Aut( B ( G w )). The fan B ( G w ) carries actions of the groupsAut( G w ) and Aut( I ( G w )), where I ( G w ) denotes the independence complex of the graph G w . Ingeneral, we have Aut( G w ) ∼ = S m × S n − , while a general description of Aut( I ( G w )) eludes theauthors. In the case n = 2, the graph G w is a star with m leaves, and I ( G w ) is a standard ( m − G w ) ∼ = Aut( I ( G w )) ∼ = S m . By Theorem 4.3, in this case both groupsare strictly smaller than the automorphism group Aut( B ( G w )) ∼ = S m × S .4.3. ∆ ,w as a flag complex. When g = 0 and w ∈ Q n ∩ (0 , n with (cid:80) w i >
2, the objects inΓ ,w are automorphism-free, and hence ∆ ,w may be realized as a simplicial complex. Given some A ⊆ { , . . . , n } with w ( A ) , w ( A c ) >
1, we put B A for a Γ ,w -object with one edge, such that onevertex supports the elements of A , and the other supports the elements of A c . A collection ofvertices { B A , . . . , B A k } spans a ( k − ,w if and only if there exists a Γ ,w -object G with precisely k edges e , . . . , 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 ∆ ,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: Theorem 4.5.
A collection { B A , . . . , B A k } of vertices of ∆ ,n spans a simplex if and only if eachpair { B A i , B A j } forms a -simplex of ∆ ,n . The analogous theorem for ∆ ,w follows from the following observation: a graph G in Γ ,n liesin Γ ,w if and only if G / { e } c lies in Γ ,w for all e ∈ E ( G ). Thus, if we are given a collection { B A , . . . , B A k } of vertices of ∆ ,w such that each pair of them spans a 1-simplex of ∆ ,w , then wecan use Theorem 4.5 to guarantee that there exists some graph G in Γ ,n such that G has precisely k edges e , . . . , e k , and such that G / { e i } c ∼ = B A i for all i . By our observation, we actually have G ∈ Ob(Γ ,w ), hence { B A , . . . , B A k } spans a simplex of ∆ ,w . As such, we have the followingcorollary of Theorem 4.5. Corollary 4.6.
The space ∆ ,w is a flag complex. In particular, we have Aut(∆ ,w ) ∼ = Aut(∆ (1)0 ,w ) , where ∆ (1)0 ,w denotes the -skeleton of ∆ ,w . UTOMORPHISMS OF TROPICAL HASSETT SPACES 17
Calculation of
Aut(∆ ,w ) for heavy/light w . We now prove the following theorem.
Theorem 4.7.
Let m, n ≥ such that m + n ≥ . Then, if w = ( ε ( m ) , ( n ) ) for ε ≤ /m , we have Aut(∆ (1)0 ,w ) ∼ = S m × S n . To prove Theorem 4.7, we will show that any automorphism of ∆ (1)0 ,w can be completely describedby its action on graphs of the form B i,j := B { i,j } , where i ∈ { , . . . m } and j ∈ { m + 1 , . . . , m + n } . Graphs of this form will be called special . Specialgraphs 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 ) denotethe number of isomorphism classes of expansions of G with precisely one more edge than G . Thenfor 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 (cid:48) ,j (cid:48) for possibly different indices i (cid:48) ∈ { , . . . m } and j (cid:48) ∈{ m + 1 , . . . , m + n } . The proof of Lemma 4.8 amounts to a somewhat tedious application of basic calculus, and canbe found in Appendix C. Establishing an analogue of this lemma for arbitrary weight vectors seemsto be the principal obstruction to determining the groups Aut(∆ ,w ) in general. Proof of Theorem 4.7.
The desired isomorphism is given by the map F : S m × S n → Aut(∆ (1)0 ,w ) , ( σ, τ ) (cid:55)→ Φ ( σ,τ ) , 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 andsurjective: F is injective. Supposing that Φ ( σ,τ ) acts as the identity on ∆ (1)0 ,w , we must show that ( σ, τ ) is theidentity permutation. We use that Φ ( σ,τ ) in particular fixes each special graph B i,j . As we areassuming m + n ≥
5, the graph B i,j has at least 3 marked points on its other endpoint. It followsthat { σ ( i ) , τ ( j ) } = { i, j } , or σ ( i ) = i and τ ( j ) = j . This demonstrates that ( σ, τ ) is the identitypermutation. F is surjective. Fix an arbitrary automorphism Ψ ∈ Aut(∆ (1)0 ,w ). Note that Ψ preserves the numberof expansions of a graph (i.e. the valence of a vertex in ∆ (1)0 ,w ). This means that for all specialgraphs B i,j , we must have that Ψ( B i,j ) is some special graph B i (cid:48) ,j (cid:48) as well. Define the permutation( σ, τ ) ∈ S m × S n via σ ( i ) := i (cid:48) and τ ( j ) := j (cid:48) . We now claim that Φ ( σ,τ ) = Ψ. Since Φ ( σ,τ ) ◦ Ψ − fixes all special graphs by definition, it suffices to check that (Φ ( σ,τ ) ◦ Ψ − )( B A ) = B A where A isan arbitrary subset of left-hand weights.For any such graph B A , we can decompose A into light and heavy weights as A = A L (cid:116) A H ,where A L ⊂ { , . . . , m } and A H ⊂ { m +1 , . . . , m + n } . Similarly we can decompose A C into disjointsets A CL and A CH . Note that the set of special graphs incident to B A is incident is then precisely { B i,j } ∪ { B i (cid:48) ,j (cid:48) } , where ( i, j ) ∈ A L × A H and ( i (cid:48) , j (cid:48) ) ∈ A CL × A CH .In general if B A is incident to special vertices { B i,j } , then A can be recovered up to complementfrom 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 onthis vertex as follows. The left-hand light indices i (cid:48) are those for which B i (cid:48) ,j is incident to B A .Similarly, the left-hand heavy indices j (cid:48) are those for which B i (cid:48) ,j (cid:48) is incident to B A for all left-hand light indices i (cid:48) . All of the weights { i (cid:48) } ∪ { j (cid:48) } either make up A or A C , so we conclude that B A isuniquely determined by its special neighbors.In summary, we know that Φ ( σ,τ ) ◦ Ψ − fixes all special neighbors of B A , and that B A is theunique one-edge graph incident to all of these special neighbors. It follows that Φ ( σ,τ ) ◦ Ψ − fixes B A as well, so Φ ( σ,τ ) = Ψ and F is surjective. (cid:3) 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 ≥ and w ∈ Q n ∩ (0 , 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 ∈ { , . . . , 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 · · · τ k where each τ i ∈ H is atransposition. First consider the case where σ = ( i , . . . , i r ) is a cycle. Then we have σ = ( i , i r )( i , i r − ) · · · ( i , i )( i , i )= ( i , σ r − ( i ))( i , σ r − ( i )) · · · ( i , σ ( i ))( i , σ ( i )) . Each transposition ( i , σ j ( i )) 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 sameargument. (cid:3) Proposition A.2.
Let w ∈ Q n ∩ (0 , n . Then the subgroup Aut( K w ) ≤ S n is generated bytranspositions.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 thereexists S ⊆ { , . . . , 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 WLOGthat i ∈ S and j / ∈ S . Write S = { (cid:96) , . . . , (cid:96) p , i } = L ∪ { i } where L = { (cid:96) , . . . , (cid:96) p } . For any natural number k ≥
0, we have σ k ∈ Aut( K w ), so we must have w ( σ k ( S )) = w ( σ k ( L )) + w σ k ( i ) ≤ , but using that L = τ ( L ) and j = σ ( i ), we have w ( σ k ( τ ( S ))) = w ( σ k ( τ ( L ))) + w σ k ( j ) = w ( σ k ( L )) + w σ k +1 ( i ) > , 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 wewanted to show. (cid:3) The following lemma allows us to symmetrize the weight data with respect to the action ofAut( K w ). Lemma A.3.
Suppose n ≥ and w ∈ Q n ∩ (0 , n . Then there exists some ˆ w ∈ Q n ∩ (0 , n suchthat UTOMORPHISMS OF TROPICAL HASSETT SPACES 19 (i) K ˆ w = K w ;(ii) if σ ∈ Aut( K ˆ w ) with σ ( i ) = j , then ˆ w i = ˆ w j .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 ˆ w obtained from w by changing both w i and w j to ( w i + w j ) / K ˆ w = K w . Indeed, suppose w ( S ) ≤ w ( S ) ≤
1. If both i, j are contained in S or S c , then w ( S ) = ˆ w ( S ), so it suffices to consider the case where i ∈ S and j / ∈ S ; write S = L ∪ { i } where i, j / ∈ L . Then ˆ w ( S ) = w ( L ) + w i + w j ≤ w ( L ) + max( w i , w j ) ≤ , since τ ∈ Aut( K w ). This shows that any S ∈ K w satisfies S ∈ K ˆ w . Conversely suppose ˆ w ( S ) ≤ 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 w ( S ) = ˆ w ( L ) + w i > . Then we must also have w ( τ ( S )) = ˆ w ( L ) + w j > . It follows that ˆ w ( S ) = ˆ w ( L ) + w i + w j ≥ ˆ w ( L ) + min( w i , w j ) > , which is a contradiction. Thus S ∈ K w , and we are done. (cid:3) 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 finiteas Aut( K w ) is finite. We have left to show that an arbitrary finite direct product of symmetricgroups can be realized in this way. Proof of Theorem 1.3.
Suppose G ∼ = k (cid:89) i =1 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 havesome vector ˆ w such that Aut( K ˆ w ) ∼ = (cid:81) k − i =1 S n i , in order to construct w such that Aut( K w ) ∼ = G .We may assume that n k > w , we say an index i ∈ { , . . . , n } is heavy in w if w i + w j > j (cid:54) = i . We say an index i is light in w if for all S ⊆ { , . . . , 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 andonly if j is also heavy, respectively light, in w . Moreover, by the previous lemma, there exists some ε > w (cid:48) is obtained from w by changing all heavy weights to 1 and light weights to ε ,then K w (cid:48) = K w .If ˆ w does not contain any heavy, respectively light, weights, then we can construct w by adding n k heavy, respectively light, weights to ˆ w , in which case we have Aut( K w ) ∼ = G . Otherwise, if ˆ w contains both heavy and light weights, we can assume all of the heavy weights are equal to 1 andthe light weights are equal to ε for some ε >
0. Also by the previous lemma we may assume thatwhenever σ ∈ Aut( K ˆ w ) satisfies σ ( i ) = j , we have ˆ w i = ˆ w j . Then we set w = ˆ w , . . . , ˆ w m , − εn k , εn k , . . . , εn k (cid:124) (cid:123)(cid:122) (cid:125) n k . We claim that Aut( K w ) ∼ = G . Indeed, suppose that τ = ( i, j ) ∈ Aut( K ˆ w ) and S ⊆ { , . . . , m + n k + 1 } satisfies w ( S ) ≤
1. Then we claim that w ( τ ( S )) ≤
1. Indeed, we have w ( τ ( S )) = w ( τ ( S ∩ { , . . . , m } )) + w ( τ ( S ∩ { m + 1 , . . . , m + n k + 1 } ))= ˆ w ( τ ( S ∩ { , . . . , m } )) + w ( S ∩ { m + 1 , . . . , m + n k + 1 } )= ˆ w ( S ∩ { , . . . , m } ) + w ( S ∩ { m + 1 , . . . , m + n k + 1 } )= w ( S ) ≤ . Conversely, if τ = ( i, j ) ∈ Aut( K w ) and i, j ≤ m , then also τ ∈ Aut( K ˆ w ). If τ = ( i, j ) ∈ Aut( K w )and i > m + 1, then we claim also j > m + 1. The weights w i for i > m + 1 are those which areequal to ε/n k , and these are the unique light weights in w . Finally, we claim that there are notranspositions τ = ( i, j ) ∈ Aut( K w ) where either i or j is equal to m + 1. This is because m + 1is the unique vertex of K w which is connected by an edge to all of the light indices, but is notconnected to any other indices: we have ˆ w i ≥ ε for all i = 1 , . . . , m .Altogether, this shows thatAut( K w ) = (cid:104) ( i, j ) | ( i, j ) ∈ Aut( K ˆ w ) or w i = w j = ε/n k (cid:105) ∼ = Aut( K ˆ w ) × S n k ∼ = G, as desired. (cid:3) Appendix B. Proof of Theorem 2.11
We now prove Theorem 2.11, restated below. We reitirate that its proof is analogous to [Kan,Proposition 3.4].
Theorem.
Let Φ ∈ Aut(∆ g,w ) . Then Φ preserves the subcomplexes V ig,w for all i ≥ . Since Aut(∆ g,w ) preserves the number of edges of each edge-labelled graph, it suffices to showthat Aut(∆ g,w ) preserves the first Betti number b ( G ) of the graph underlying a simplex [ G , τ ].This is clear when g = 0, for in this case we have b ( G ) = 0 for all w -stable graphs G . Now fix g ≥
1, and k such that 1 ≤ k ≤ g . Put R k for the unique (up to isomorphism) Γ g,w -object withone vertex and k loops. Each R k has a unique edge-labelling up to the action of Aut E ( R k ) ∼ = S k .We put [ R k ] ∈ ∆ g,w [ k −
1] for the corresponding simplex of ∆ g,w . Then given a simplex [ G , τ ], wehave that b ( G ) ≥ k if and only if [ G , τ ] has [ R k ] as a face. Thus, to prove that Aut(∆ g,w ) fixesthe first Betti number of each graph, it suffices to prove that it fixes each [ R k ]. Finally, each [ R k ]is a face of [ R g ], so it is enough just to show that [ R g ] is fixed. We prove this in intermediate steps,the first being that the vertex [ R ] ∈ ∆ g,w [0] is fixed. Proposition B.1.
Suppose g ≥ . Then for any Φ ∈ Aut(∆ g,w ) , we have Φ[ R ] = [ R ] .Proof. We say a graph G of Γ g,w is maximal if the only graphs admitting morphisms to G arethemselves isomorphic to G . Edge-labellings of maximal graphs correspond to facets of ∆ g,w ,where a facet of a symmetric ∆-complex is a simplex which is not a proper face of any othersimplex. An automorphism of a symmetric ∆-complex must permute the d -dimensional facetsamongst themselves. We claim that [ R ] ∈ ∆ g,w [0] is the unique vertex which is a face of all facetsof ∆ g,w . Graph-theoretically, this is equivalent to the statement that R is the unique graph in Γ g,w which has one edge and which admits a morphism from all maximal graphs. Indeed, any maximalgraph G satisfies b ( G ) = g , and so must have at least one cycle. We thus get a morphism G → R by contracting all edges except some fixed edge which is contained in a cycle of G . To see that R is the unique graph with these properties, it suffices to exhibit a maximal graph G of Γ g,w , suchthat if there exists a morphism G → H with | E ( H ) | = 1, then H ∼ = R . When g = 1, such a graph G can be constructed by taking an n -cycle and putting one marking at each vertex. For each g ≥ G such that: • G is trivalent; UTOMORPHISMS OF TROPICAL HASSETT SPACES 21 • b ( G ) = g ; • G has no bridgeswhere a bridge of a graph G is a non-loop edge which is not contained in any cycles (see Figure3 in [Kan] for an example of such a graph in general). Then the necessary Γ g,w object G can beconstructed by choosing n points on the interiors of edges of G , and putting a vertex supporting amarking at each chosen point. The graph G cannot contract to a graph which has a bridge, so theonly graph with one edge that it contracts to is R . (cid:3) To prove that the simplex [ R g ] is preserved, we first preserve that bridge indices are preserved,as in the following lemma. Lemma B.2.
Let [ G , τ ] ∈ ∆ g,w [ p ] , and put B G τ ⊆ [ p ] for the indices of bridges of G . Suppose that Φ ∈ Aut(∆ g,w ) and [ G (cid:48) , τ (cid:48) ] = Φ[ G , τ ] . Then B G τ = B G (cid:48) τ (cid:48) . Proof.
Given a simplex [ G , τ ] ∈ ∆ g,w [ p ] 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 thecommutativity of diagram 2.7, it can be shown that for any automorphism Φ of ∆ g,n , we haveΦ d S [ G , τ ] = d S Φ[ G , τ ]. With this notation in place, we can characterize B G τ as follows: B G τ = { i ∈ [ p ] | d [ p ] (cid:114) { i } [ G , τ ] (cid:54) = [ R ] } . That is, an edge e is a bridge of G if and only if upon contracting all edges in G besides e , we donot get a loop. The lemma now follows from this description of B G τ and Proposition B.1. (cid:3) We can now prove that automorphisms preserve the simplex [ R g ]. Proposition B.3.
Let g ≥ and suppose Φ ∈ Aut(∆ g,w ) . Then Φ[ R g ] = [ R g ] .Proof. Suppose G is a maximal graph of Γ g,w , with the property that every bridge of G is eithera loop or a bridge (it is straightforward to construct examples of such G for all g ≥ w ). Let τ : E ( G ) → [ p ] be any edge-labelling of G , and put [ G (cid:48) , τ (cid:48) ] = Φ[ G , τ ]. Then weclaim G (cid:48) also has the property that all of its bridges are either loops or bridges. Indeed, G (cid:48) mustalso be maximal, so b ( G (cid:48) ) = b ( G ) = g , and hence we have | V ( G (cid:48) ) | = | V ( G ) | . By Lemma B.2, G (cid:48) has the same number of bridges as G , and if we set B = B G τ ⊂ [ p ], then B indexes the bridgesin both G and G (cid:48) . Since bridges are contained in all spanning trees, the edges indexed by B in G (cid:48) must be contained in some spanning tree of G (cid:48) . On the other hand, we know the edges indexedby B in G form a spanning tree of G . Since G and G (cid:48) have the same number of vertices, theyhave the same number of edges in a spanning tree. Therefore the edges indexed by B in G (cid:48) form aspanning tree. Whenever we contract a spanning tree in a Γ g,w -object of first Betti number g , theresulting graph is R g . In particular, we haveΦ[ R g ] = Φ d B [ G , τ ] = d B [ G (cid:48) , τ (cid:48) ] = [ R g ] , and the proof is complete. (cid:3) 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
We restate the lemma for convenience:
Lemma.
Consider the same hypotheses as Theorem 4.7. For a graph G , let exp( G ) denote thenumber of isomorphism classes of expansions of G with precisely one more edge than G . Then forall 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 (cid:48) ,j (cid:48) for possibly different indices i (cid:48) ∈ { , . . . m } and j (cid:48) ∈{ 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 asoccupying 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 bijectivecorrespondence with subsets S of A such that w ( S ) > | A (cid:114) S | >
0. There are 2 x − y − ( x − y ) − 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 heavyweight, minus the subset A itself. Repeating the counting argument on the other side, we concludethat exp( B A ) = 2 x − y − ( x − y ) − ( m + n ) − x − m − y − [( m + n − x ) − ( m − y )] −
1= 2 x − y + 2 m + n − x − m − y − − n expansions total.It therefore suffices to maximize f ( x, y ) := 2 x − y + 2 ( m + n ) − x − m − y over a domain that includes all permissible integer values of ( x, y ). Such a domain is determinedby the three inequalities 2 ≤ x ≤ ( m + n ) −
2, 0 ≤ y ≤ m , and 1 ≤ ( x − y ) ≤ n −
1; see Figure 10. (2 ,
1) ( m + 1 , m ) ( m + n − , m )( m + n − , m − n − , m + n , m )(2, 0) ( m + n , Figure 10.
The domain of f ( x, y )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 numberof light weights on either vertex cannot exceed m , the total number of light weights. Finally, theremust 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 − n = 2separately from n ≥ n = . As there are exactly two heavy weights, each vertex of B A supports one of them. In otherwords, we have y = x −
1. We are now maximizing the function f ( x, x −
1) = 2 x − x − + 2 ( m +2) − x − m − ( x − = 2 x − + 2 m − x +1 . UTOMORPHISMS OF TROPICAL HASSETT SPACES 23 over the interval 2 ≤ x ≤ m . As the second derivative d dx (2 x − + 2 m − x +1 ) = 2 − x − log (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 ≥ . First, we look for critical points on the interior of the region. We compute the gradient as ∇ f = (cid:104) log(2)2 x − m + n − x , − y log(2)(2 m − y ) (cid:105) . Setting the partial derivatives equal to 0, we findthat there is one critical point located at (( m + n ) / , m/ f over the boundary. Note that there is a symmetry originating from exchangingthe two vertices of B A ; symbolically, this is the involution ( x, y ) (cid:55)→ ( m + n − x, m − y ). Thereforeit suffices to optimize f over only half of the boundary, i.e. only over the left-hand equalities.Specifically, we consider restricting f to the following the three boundary segments: f (2 , y ) , for 0 ≤ y ≤ ,f ( x, , for 2 ≤ x ≤ n − f (1 + y, y ) , for 1 ≤ y ≤ m. We first look for critical points on the interiors of these segments, and second consider the valuesof f at their endpoints. • First, note that f (2 , y ) = 2 m + n − − m − y − 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 − m + 2 x − x = ( m + n ) /
2. This point is interior when 2 < ( m + n ) / m + n ) / < n −
1. Thatis, there is a critical point interior to this edge whenever n > m + 2. • Finally, note that f (1 + y, y ) = 2 m + n − y − − m − y + 2 y has a critical point when 2 m + 2 y =2 m + n − . Since m (cid:54) = m + n − y = m + n − y (cid:54) = m all of the exponents in this equation are distinct. Thus there is no integersolution by the uniqueness of binary representations. In case 2 y = m the equation reducesto m + 1 = m − n −
1, or n = 2. As we are assuming n ≥
3, this edge contains no criticalpoints.We now consider the function values at the interior critical point, the point on the boundaryedge interior that is sometimes a critical point, and three of the six vertices: f (( m + n ) / , m/
2) = 2 ( m + n ) / − m/ f (( m + n ) / ,
0) = 2 ( m + n ) / − − m f (2 ,
0) = 2 m + n − − m + 3 f (2 ,
1) = 2 m + n − − m − + 2 f ( n − ,
0) = 2 m + 2 n − − . We claim that f (2 ,
1) is at least as big as all of these values. It suffices to check the followingfour inequalities: • f (( m + n ) / , m/ < f (2 , ( m + n ) / − m/ < m + n − − m − + 2 , or equivalently that2 m − − m/ < m + n − − ( m + n ) / + 2 . It suffices to show that the function g ( x ) = 2 x − ( x +3) / is non-decreasing in x , for x ≥
2. Indeed we have g (cid:48) ( x ) = log 2 · x − log 22 · ( x +3) / = log 2 · x − log 2 · / · x/ > x >
1, proving the claim. • f (( m + n ) / , < f (2 , ( m + n ) / < m + n − + 2 m − + 3 / . For m + n ≥ ( m + n ) / ≤ m + n − < m + n − + 3 / , proving the claim in every case except when m + n = 5. In that case, the inequality becomes2 m > √ − ≈ . , which holds since m ≥ • f (2 , < f (2 , − m < − m − , which is true for 1 < m − , i.e.for all m > • f ( n − , < f (2 , m − + 2 m + 2 n − < m + n − + 3 , or (2 m − n − > . As we have m >
1, the inequality reduces to 2 n >
6, or n > . ... . But n ≥
3, so thisinequality holds. (cid:3)
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Department of Mathematics, Brown University, Providence, RI 02906
Email address , S. Freedman: [email protected]
Email address , J. Hlavinka: [email protected]
Email address , S. Kannan:, S. Kannan: