6.1 Metric graphs with edge-multiplicities
Definition 6.1.1. Let Γ be a metric graph, and (G, l) a model of Γ. We call a function m : E(G) → Z>0 an edge-multiplicity function on G. 1 is the edge-multiplicity function assigning multiplicity one to all edges and called a trivial edge-multiplicity function. Two triplets (G, l, m) and (G′, l′, m′) are said to be isomorphic if there exists an isomorphism betweenGandG′keeping the length and the multiplicity of each edge. We define Isom(G,l)(Γ) as the subset of the isometry transformation group Isom(Γ) of Γ whose element keeps the length of each edge ofG. We set Isom(G,l,m) as the subset of Isom(G,l)(Γ) whose each element keeps the multiplicity of each edge of G.
Definition 6.1.2 (Subdivision of models). Let Γ be a metric graph, and (G, l), (G′, l′) models of Γ. (G, l) is said to be a subdivision of (G′, l′) and written as (G, l) ≻ (G′, l′) if V(G′) is a subset of V(G).
Definition 6.1.3. Let Γ be a metric graph, and (G, l) ≻ (G′, l′) models of Γ. A triplet (G, l, m) is said to be asubdivisionof a triplet (G′, l′, m′) and written as (G, l, m)≻(G′, l′, m′) if for any e′ ∈ E(G′) and ei ∈E(G) such that e′ =e1⊔ · · · ⊔en, m′(e′) divides allm(ei). In particular, ifm′(e′) and allm(ei) equals, then (G′, l′, m′) is said to be atrivialsubdivision of (G, l, m) and then (G′, l′, m′) is denoted by (G′, l′, m).
Definition 6.1.4. For a quadruplet (Γ, G, l, m), the metric graph with an edge-multiplicity, denoted by Γm, is defined by the pair of metric graphΓ and msuch that we can choose only models (G′, l′) ≺(G, l) of Γ. The word “a point x on Γm” means that x∈ Γ. The genus of Γm is the genus of Γ.
Definition 6.1.5. Let Γm be a metric graph with an edge-multiplicity. Div(Γm) is defined by Div(Γ) and an element of Div(Γm) is called adivisoronΓm. Thecanonical divisoronΓm is the canonical divisor on Γ. We define Rat(Γm) as Rat(Γ). We call an element of Rat(Γm) a rational function onΓm.
Note that for an edge e of G and f ∈Rat(Γm), f has different finite slopes on e since f may have plural pieces.
For a metric graph with an edge-multiplicity, we use same terms and notations for the underlying metric graph.
LetΓm′′ be not a singleton and x a point on Γm. The morphism φm, m′ is harmonic at x if for any edge e1 of G adjacent to x, m(e1) devides m′(φm, m′(e1)) and the number
degm, mx ′(φm, m′) := ∑
x∈h7→h′
m′(φm, m′(e))
m(e) ·degh(φm, m′)
is independent of the choice of half-edge h′ emanating fromφm, m′(x), whereh is a connected component of the inverse image ofh′ byφm, m′ containingxande is the edge ofGcontaining h. The morphism φm, m′ is harmonicif it is harmonic at all points on Γm. For a point x′ on Γm′′,
degm, m′(φm, m′) := ∑
x7→x′
degm, mx ′(φm, m′)
is said the degreeof φm, m′, where xis an element of the inverse image ofx′ byφm, m′. If Γm′ ′
is a singleton and Γm is not a singleton, for any point x onΓm, we define degm, mx ′(φm, m′) as zero so that we regardφm, m′ as a harmonic morphism of degree zero. If bothΓm andΓm′′ are singletons, we regard φm, m′ as a harmonic morphism which can have any number of degree.
Lemma 6.2.3. ∑
x7→x′degm, mx ′(φm, m′)is independent of the choice of a point x′ on Γ′. Proof. It is sufficient to check that for any vertex of G′, the sum is same. Let x′1 and x′2 be vertices of G′ both adjacent to an edge e′ of G′. Let h′1 be the half-edge of x′1 contained in e′. Then
∑
x17→x′1
degm, mx1 ′(φm, m′) = ∑
x17→x′1
∑
x1∈h17→h′1
degm, mh ′
1 (φm, m′)
= ∑
x17→x′1
( ∑
x1∈e17→e′
degm, me ′
1 (φm, m′) )
= ∑
e7→e′
degm, me ′(φm, m′).
Similarly, ∑
x27→x′2
degm, mx2 ′(φm, m′) = ∑
e7→e′
degm, me ′(φm, m′).
The collection of metric graphs with edge-multiplicities together with harmonic mor-phisms between them forms a category.
Definition 6.2.4. Let φm, m′ : Γm → Γm′′ be a finite harmonic morphism of metric graphs with edge-multiplicities. For f in Rat(Γm), thepush-forward off is the function (φm, m′)∗f : Γm′′ →R∪ {±∞}defined by
(φm, m′)∗f(x′) := ∑
x∈Γm
φm, m′(x)=x′
degm, mx ′(φm, m′)·f(x).
The pull-back of f′ in Rat(Γm′ ′) is the function (φm, m′)∗f′ : Γm → R ∪ {±∞} defined by (φm, m′)∗f′ :=f′◦φm, m′. We define the push-forward homomorphism on divisors (φm, m′)∗ : Div(Γm)→Div(Γm′ ′) by homomorphism
(φm, m′)∗(D) := ∑
x∈Γm
D(x)·φm, m′(x).
The pull-back homomorphism on divisors (φm, m′)∗ : Div(Γm′′)→Div(Γm) is defined to be (φm, m′)∗(D′) := ∑
x∈Γm
degm, mx ′(φm, m′)·D′(φm, m′(x))·x.
Remark 6.2.5. We need not to assume that φm, m′ is finite to define pull-backs of rational functions and divisors.
Proposition 6.2.6. For any divisorsD on Γm and D′ on Γm′′, deg((φm, m′)∗(D)) = deg(D) and deg((φm, m′)∗(D′)) = degm, m′(φm, m′) hold.
Proof. The first equation holds obviously.
Let x′ be a point on Γm′ ′. Since ∑
x7→x′((φm, m′)∗(D′))(x) = ∑
x7→x′degm, mx ′(φm, m′) · D′(x′) = degm, m′(φm, m′)·D′(x′), we have the second equation.
Definition 6.2.7. Let φm, m′ : Γm → Γm′′ be a finite harmonic morphism of metric graphs with edge-multiplicities. For a rational function f on Γm other that −∞, we define the number
divm, m′(f) := ∑
x∈Γm
∑
x∈e∈E(G)
m′(φm, m′(e))
m(e) ·(the outgoing slope of f one at x)·x
and call it the principal divisor with edge-multiplicities m and m′ defined by f.
Proposition 6.2.8. For any rational functionsf onΓm andf′ onΓm′′ both other than−∞, (φm, m′)∗(divm, m′f) = div((φm, m′)∗(f)) and (φm, m′)(div(f′)) = divm, m′((φm, m′)∗f′) hold.
Proof. Let us write φm, m′ as φ simply. We may break Γm and Γm′ ′ into sets S and S′ of segments along which f and φ∗f, respectively, are linear and such that each segment s∈ S is mapped linearly to some s′ ∈S′. Then at any point x′ onΓm′ , we have
φ∗(divm, m′(f))(x′) = ∑
x∈Γm
x7→x′
divm, m′(f)(x) = ∑
x∈φ−1(x′)
∑
s=xy∈S
m′(φ(s))
m(s) · f(y)−f(x) l(s)
and
div(φ∗f)(x′)
= ∑
s′=x′y′∈S′
(φ∗f)(y′)−(φ∗f)(x′) l′(s′)
= ∑
s′=x′y′∈S′
∑
y∈Γm
φ(y)=y′
( ∑
y∈s7→s′
m′(s′)
m(s) · l′(s′) l(s)
)
f(y)− ∑
x∈Γm
φ(x)=x′
( ∑
x∈s7→s′
m′(s′)
m(s) · l′(s′) l(s)
) f(x)
· 1 l′(s′)
= ∑
s′=x′y′∈S′
∑
y∈Γm
φ(y)=y′
( ∑
y∈s7→s′
m′(s′) m(s) · 1
l(s) )
f(y)− ∑
x∈Γm
φ(x)=x′
( ∑
x∈s7→s′
m′(s′) m(s) · 1
l(s) )
f(x)
= ∑
s′=x′y′∈S′
∑
s=xy∈S φ(s)=s′
(m′(s′) m(s) · f(y)
l(s) − m′(s′)
m(s) · f(x) l(s)
)
= ∑
x∈φ−1(x′)
∑
s=xy∈S
m′(φ(s))
m(s) · f(y)−f(x) l(s) .
Let us assume that Γm and Γm′ ′ are broken intoS1 and S1′ of segments along which φ∗f′ and f′, respectively, have the same conditions as that of S and S′. Then for any point x on Γm, we have
(φ∗(div(f′))) (x) = degm, mx ′(φm, m′)·(div(f′)(φ(x)))
= degm, mx ′(φm, m′)·
∑
s′=φ(x)y′∈S′
f′(y′)−f′(φ(x)) l′(s′)
= ∑
s′=φ(x)y′∈S′
degm, mx ′(φm, m′)· f′(y′)−f′(φ(x)) l′(s′)
= ∑
s′=φ(x)y′∈S′
∑
s=xy7→s′
m′(s′)
m(s) · l′(s′)
l(s) ·f′(y′)−f′(φ(x)) l′(s′)
= ∑
s′=φ(x)y′∈S′
∑
s=xy7→s′
m′(s′)
m(s) · f′(y′)−f′(φ(x)) l(s)
= ∑
s=xy∈S
m′(s′)
m(s) · f′(φ(y))−f′(φ(x)) l(s)
= ∑
s=xy∈S
m′(s′)
m(s) · (φ∗f′)(y)−(φ∗f′)(x) l(s)
= (divm, m′(φ∗(f′)))(x).
Definition 6.2.9. Let φm, m′ : Γm → Γm′′ be a map between metric graphs with edge-multiplicities m and m′ and let K be a finite group. φm, m′ is a K-Galois covering on Γm′ ′
if φm, m′ is a finite harmonic morphism of metric graphs with edge-multiplicities, |K| = degm, m′(φm, m′) and K acts on transitively on every fibre and K keeps edge-multiplicities.
Remark 6.2.10. Ifφm, m′ :Γm →Γm′ ′ isK-Galois, then there exists a group homomorphism K →Isom(G,l,m) for a model (G, l) ofΓm.
Appendix
In this appendix, we show Proposition 7. For this, we define some concepts and prove lemmas and propositions (cf. [3]).
Anelementary tropical modificationof a metric graphΓ0 is a metric graphΓ = [0,+∞]∪ Γ0 obtained fromΓ0 by attaching the segment [0,+∞] to Γ0 in such a way that 0 ∈[0,+∞] gets identified with a finite point p on Γ0. A metric graph Γ obtained from a metric graph Γ0 by a finite sequence of elementary tropical modifications is called a tropical modification of Γ0. Tropical modifications generate an equivalence relation on the set of metric graphs.
This equivalence relation is called tropical equivalence.
IfΓ is a tropical modification ofΓ0, then there is a natural strong deformation retraction map τ : Γ → Γ0 which is the identity on Γ0 and contracts each connected component of Γ\Γ0 to the unique point inΓ0 lying in the topological closure of that component. The map τ is a non-finite harmonic morphism of metric graphs.
A metric graph Γ is minimal if there exist no inverse elementary tropical modifications of Γ.
Proposition 1. Let Γ be a metric graph. Γ is minimal if and only if for any point x on Γ, val(x)̸= 1.
Proof. (only if part) We prove the contraposition. Suppose there exists a point xon Γ with valence one. As Γ is a metric graph, there exists an open neighborhood of x. Since x has valence one, there exists a closed segmentsinU withxas an endpoint. LetΓ′ be the closure of Γ \s. ThenΓ′ is an inverse tropical modification ofΓ. Therefore Γ is not minimal.
(if part) By the definition of minimality of metric graphs, the statement obviously holds.
Lemma 2. Let Γ be a metric graph of genus ≥1. Then there exist a minimal metric graph Γ′ tropically equivalent to Γ and a retraction π :Γ →Γ′.
Proof. Let V1 be the set of one valent points on Γ. Fix any point x1 ∈ V1. Let e1 be the edge of Γ adjacent to x1 and y1 the vertex of e1 different fromx1. Let fΓ1 be the elementary tropical modification of Γ obtained by attaching [0,∞] atx1 toΓ and Γ1 the metric graph obtained by retracting e◦1∪ {x1} to y1. Let τ1 : Γ → Γ1 be the natural retraction. Γ1 is a tropical modification of Γ since Γ1 is an elementary tropical modification of fΓ1.
Let V2 be the set of one valent points on Γ1. |V2| is less than |V1|. Let x2 be a point in V2, e2 the edge of Γ1 adjacent to x2 and y2 be the vertex of e2 different from x2. LetΓ2 be the metric graph obtained by retracting e◦2∪ {x2} toy2. By the same reason,Γ2 is a tropical modification of Γ. Let τ2 :Γ →Γ2 be the natural retraction.
Repeating this operation until the set Vd of one valent points of Γd becomes empty, we obtain a finite sequence of metric graphs Γ1, Γ2, . . . , Γd and a finite sequence of retractions τ1, τ2, . . . , τd. The composition τd◦. . .◦τ2◦τ1 is a retraction from Γ toΓd and Γd is minimal and tropically equivalent to Γ.
Proposition 3 (Universal mapping property). For any metric graph Γ1 of genus ≥1, any minimal metric graph Γmin′ and any harmonic morphism f :Γ1 →Γmin′ , there exists a unique harmonic morphism ψ :Γ1,min →Γmin′ such that φ=ψ◦π, where Γ1,min is a minimal metric graph tropically equivalent to Γ1 and π :Γ1 →Γ1,min is a retraction.
Proof. Sinceπis a retraction, there exists an injectionι:Γ1,min →Γ1 such thatπ◦ι= idΓ1,min. Letψ :=φ◦ι. Then φ=ψ◦π. Indeed, for any 1-valent pointv onΓ1, 1 = val(v)<val(φ(v)) as Γmin is minimal. Therefore there exists a half-edge h′ of φ(v) such that the closure of the inverse image of h′ by φ dose not contain v, that is, ∑
h7→h′,¯h̸∋vdegh(φ) = 0. Since φ is harmonic at v, the degree of φatv is zero, so φdegenerates on the edge containing v.
By Lemma 2 and Proposition 3, we have the following corollary.
Corollary 4. Let Γ be a metric graph of genus ≥ 1. Then there exist a unique minimal metric graph Γ′ tropically equivalent to Γ and a unique retraction π :Γ →Γ′.
Lemma 5. Let φ: Γ → Γ′ be a harmonic morphism of metric graphs and let (G, l) (resp.
(G′, l′)) be the loopless model ofΓ (resp. Γ′) corresponding toφ. For any vertex v of Gsuch thatdegv(φ)̸= 0, val(v)−A≥val(φ(v))holds, where Ais the number of edges of Gadjacent to v which degenerate by φ.
Proof. Let us assume that there exists a vertexv ofGsuch that degv(φ)̸= 0 and val(v)−A <
val(φ(v)). There exists an edgee′ of G′ adjacent toφ(v) whose inverse image by φdoes not contain an edge adjacent to v. Then ∑
e7→e′,v∈edege(φ) = 0. Therefore degv(φ) = 0 since φ is harmonic at v. This is a contradiction.
For such a vertexv in Lemma 5, let valφ(v) := val(v)−A.
Lemma 6. Let φ: Γ →Γ′ be a harmonic morphism of metric graphs. If deg(φ) ≥1, φ is surjective.
Proof. If φ is not surjective, then there exists a point x′ on ∂Im(φ). Let h′ be a half-edge of x′ whose inverse image by φ is empty. Then, for any point x in φ−1(x′), degx(φ) is zero.
Thus contradicts deg(φ)≥1.
Proposition 7. Let φ:Γ → Γ′ be a harmonic morphism of metric graphs. If deg(φ)≥ 1, then the genus g(Γ) of Γ is not less than the genus g(Γ′) of Γ′.
Proof. By Lemma 6, it is enough to examine the restriction of φ onΓmin. Let (Gmin, l) and (G′, l′) be models ofΓmin and φ(Γmin) corresponding to φ, respectively. By the handshaking lemma and Lemma 5 and Lemma 6,
|E(Gmin)| − |E(G′)|
= 1
2
∑
v∈V(Gmin)
val(v)− ∑
v′∈V(G′)
val(v′)
= |V(Gmin)|+ ∑
v∈V(Gmin) val(v)≥2
val(v)−2
2 − |V(G′)|+ ∑
v′∈V(G′) val(v′)=1
val(v′)
2 − ∑
v′∈V(G′) val(v′)≥2
val(v′)−2 2 and
∑
v∈V(Gmin) val(v)≥2
val(v)−2
2 + ∑
v′∈V(G′) val(v′)=1
val(v′)
2 − ∑
v′∈V(G′) val(v′)≥2
val(v′)−2 2
≥ ∑
v∈V(Gmin) val(v)≥2
(val(v)−2
2 − val(φ(v))−2 2
)
≥0
hold. Thus,
g(Γ)−g(Γ′) = g(Γmin)−g(φ(Γmin))
= (|E(Gmin)| − |V(Gmin)|+ 1)−(|E(G′)| − |V(G′)|+ 1)
= (|E(Gmin)| − |E(G′)|)−(|V(Gmin)| − |V(G′)|)≥0, i.e. g(Γ)≥g(Γ′) holds.
Remark 8. Graph version of Proposition 7 can be proven easily by using Riemann–Hurwitz formula for graphs in [5]. However, since we does not prove Riemann–Hurwitz formula for metric graphs, we give another proof for Proposition 7.
Acknowledgement
I would like to express my gratitude for my advisor, Masanori Kobayashi for supervising this research and generously giving a lot of advice. I also thank my laboratory members, Yasuhito Nakajima, Kohei Takagi and Yuki Kageyama, for being able to have plenty of discussions, and my family for supporting my fruitful university life.
References
[1] 秋庭 芳江, 非特異トロピカル平面曲線の補空間について, 首都大学東京修士論文, 2015.