1·1. Finite graphs and harmonic morphisms

We use a modified version of Serre’s definition of a graph [ Reference SerreSer80 ] that allows for legs, which are a type of extremal edge with no end vertex.

The set X(G) is the union of the vertices V(G) and half-edges H(G) of the graph G, where V(G) is the image of r and $H(G)=X(G)\backslash V(G)$ is the complement. The involution $\iota$ preserves H(G) and partitions it into orbits of sizes 1 and 2; we call these respectively the legs and edges of G and denote the corresponding sets by L(G) and E(G). The root map assigns one root vertex to each leg and two root vertices to each edge (each vertex is rooted at itself). A loop is an edge whose root vertices coincide. An orientation on G is a choice of order (h,h ^′) on each edge $e=\{h,h'\}$ of G and defines source and target maps $s,t\;:\;E(G)\to V(G)$ by $s(e)=r(h)$ and $t(e)=r(h')$ . We note that a leg does not have a vertex at its free end and is thus distinct from an extremal edge, and that legs do not require orienting.

Graphs with legs naturally appear in tropical moduli problems, where a leg represents the tropicalisation of a marked point. An extremal edge, on the other hand, represents an irreducible component attached to the rest of the curve at a single node.

The tangent space $T_v G$ and valency $\textrm{val} (v)$ of a vertex $v\in V(G)$ are defined by

\begin{align*} T_vG=\big\{h\in H(G)\;:\;r(h)=v\big\} \quad\textrm{ and } \quad\textrm{val}(v)=|T_v G|,\end{align*}

so that a leg is counted once for valency, while a loop is counted twice.

A morphism of graphs $f\;:\;G'\to G$ , is a set map $f\;:\;X(G')\to X(G)$ that commutes with the root and involution maps and that sends vertices to vertices, edges to edges, and legs to legs. By abuse of notation, we denote by f the corresponding maps on the vertices, half-edges, edges, and legs. We note that our graph morphisms are finite and do not allow edges or legs to contract to vertices. Non-finite morphisms are relevant to tropical geometry, but do not occur as quotients by finite group actions; so we do not consider them.

Let G and G ^′ be graphs. A harmonic morphism $(f,d_f)$ consists of a graph morphism $f\;:\;G'\to G$ and a degree assignment $d_f\;:\;X(G')\to \mathbb{Z}_{>0}$ such that $d_f(h'_{1})=d_f(h'_{2})$ for each edge $e'=\{h'_{1},h'_{2}\}\in E(G')$ (a quantity that we denote by $d_f(e')$ ), and such that

(2) \begin{equation}d_{f}(v')=\sum_{h'\in T_{v'} G'\cap f^{-1}(h)}d_{f}(h')\end{equation}

for every $v'\in V(G')$ and every $h\in T_{f(v)}G$ . In particular, the quantity appearing on the right-hand side of (2) does not depend on the choice of $h\in T_{f(v)}G$ . The degree $d_f$ is also called the dilation factor of f. If G is connected, then the global degree of f is defined as

\begin{align*} \deg(f)=\sum_{v'\in f^{-1}(v)}d_f(v')=\sum_{e'\in f^{-1}(e)}d_f(e')=\sum_{l'\in f^{-1}(l)}d_f(l')\end{align*}

for any choice of $v\in V(G)$ , $e\in E(G)$ or $l\in L(G)$ . A harmonic morphism $(f,d_f)$ is called free if $d_f(x)=1$ for all $x\in X(G)$ ; a free harmonic morphism is a covering space in the topological sense.

1·2. Group quotients and harmonic Galois covers

An automorphism of a graph G is a morphism $f\;:\;G\to G$ that has an inverse; such a morphism can be made harmonic by setting $d_f=1$ everywhere. A priori, a non-trivial automorphism may flip edges, in other words exchange the two half-edges making up an edge. Such automorphisms do not give rise to a quotient, however, since we do not allow an edge to map to a leg. Hence we exclude them from consideration.

Given a $\mathfrak{G}$ -action on a graph G, we can naturally form the quotient graph $G/\mathfrak{G}$ in such a way that the quotient map $f\;:\;G\to G/\mathfrak{G}$ is harmonic of degree $|\mathfrak{G}|$ .

Definition 1·3. Let G be a graph and let $\mathfrak{G}$ be a finite group. Given a $\mathfrak{G}$ -action on G, we define the quotient graph $G/\mathfrak{G}$ by setting $X(G/\mathfrak{G})=X(G)/\mathfrak{G}$ . The root and involution maps on G are $\mathfrak{G}$ -invariant and descend to $X(G/\mathfrak{G})$ . It is clear that $V(G/\mathfrak{G})=V(G)/\mathfrak{G}$ and $H(G/\mathfrak{G})=H(G)/\mathfrak{G}$ , and by assumption the $\mathfrak{G}$ -action does not identify the two half-edges of any edge of G. Therefore $E(G/\mathfrak{G})=E(G)/\mathfrak{G}$ and $L(G/\mathfrak{G})=L(G)/\mathfrak{G}$ , and the quotient map

\begin{align*} f\;:\;G\longrightarrow G/\mathfrak{G}\end{align*}

is a finite morphism. By the orbit-stabiliser theorem, we can promote f to a harmonic morphism of global degree $\deg(f)=|\mathfrak{G}|$ by setting $d_f(x)=|\mathfrak{G}_x|$ , where $\mathfrak{G}_x$ is the stabiliser subgroup of $x\in X(G)$ .

We now define a harmonic Galois cover of a graph to be any harmonic morphism obtained in this way.

Let $f\;:\;G'\to G$ be a harmonic $\mathfrak{G}$ -cover, and pick a vertex or half-edge $x\in X(G)$ . The group $\mathfrak{G}$ acts transitively on the fiber $f^{-1}(x)$ , so we can identify the latter with $\mathfrak{G}/\mathfrak{G}_{x'}$ , where $\mathfrak{G}_{x'}$ is the stabiliser of some $x'\in f^{-1}(x)$ . On the other hand, for any $x',x''\in f^{-1}(x)$ we have $|\mathfrak{G}_{x'}|=|\mathfrak{G}_{x''}|$ and $d_f(x')=d_f(x'')$ . Since the degrees of f on the fiber $f^{-1}(x)$ add up to $\deg f=|\mathfrak{G}|$ , it follows that $d_f(x')=|\mathfrak{G}_{x'}|$ for any $x'\in X(G')$ . It follows that f is the quotient morphism $G'\to G'/\mathfrak{G}$ .

1·3. Metric graphs

Let G be a graph and let $\ell\;:\;E(G)\to \mathbb{R}_{>0}$ be an assignment of positive real lengths to the edges of G. The pair $(G,\ell)$ , known as a model for G, determines a metric graph $\Gamma$ by gluing a closed line segment $[0,\ell(e)]$ for each edge $e\in E(G)$ and an open infinite interval $[0,\infty)$ for each leg $l\in L(G)$ in accordance with the structure of G. We equip $\Gamma$ with the shortest-path metric. We note that the set of legs does not depend on the choice of model and that the metric graph $\Gamma$ is compact if and only if G has no legs.

A model $(G,\ell)$ of a metric graph $\Gamma$ is called simple if G has no loops or multi-edges. Given a simple model $(G,\ell)$ for $\Gamma$ , we define the star cover

\begin{align*} \mathcal{U}(G)=\{U_v\}_{v\in V(G)}\cup \{U_l\}_{l\in L(G)}\end{align*}

of $\Gamma$ as follows. For each leg $l\in L(G)$ , let $U_l\subset \Gamma$ be the interior of the corresponding infinite segment in $\Gamma$ . For each vertex $v\in V(G)$ , let $U_v\subset \Gamma$ be the union of v and the interiors of all legs and edges incident to v. The distinct $U_l$ have empty intersections, and $U_l\cap U_v=U_l$ if l is rooted at v and is empty otherwise. Finally, for distinct vertices v and w, the intersection $U_v\cap U_w$ is either the open edge connecting v and w, if there is such an edge, or is empty otherwise. Hence each element of $\mathcal{U}(G)$ is contractible, pairwise intersections are open intervals or empty, and all triple intersections are empty, making the star cover convenient for cohomological calculations.

We now define harmonic morphisms and Galois covers of metric graphs. Let $\Gamma$ and $\Gamma'$ be metric graphs with models G and G ^′, respectively. Let $f\;:\;G'\to G$ be a harmonic morphism of graphs satisfying the condition

(3) \begin{equation} \ell\big(f(e')\big)=d_f(e')\ell(e')\end{equation}

for all $e'\in E(G')$ . We define an associated continuous map $\phi\;:\;\Gamma'\to \Gamma$ of metric graphs by mapping vertices to vertices, edges to edges, and legs to legs according to f. Along each edge and leg of $\Gamma'$ , the map $\phi$ is linear with positive integer slope, or dilation factor, given by the degree $d_f$ (which we also denote $d_{\phi}$ ). Condition (3) ensures that $\phi$ is continuous, and no condition is required along the infinite legs.

A harmonic morphism of metric graphs $\phi\;:\;\Gamma'\to \Gamma$ is any continuous, piecewise-linear map obtained in this manner, with nonzero integer slopes given by the degree function $d_f$ of a harmonic morphism of graphs $f\;:\;G'\to G$ (and thus satisfying the balancing condition (2) at each vertex of $\Gamma'$ ). This definition is equivalent to requiring that $\phi$ pulls back harmonic functions on $\Gamma$ to harmonic functions on $\Gamma'$ . We refer to the datum $(G,G',f\;:\;G'\to G,d_f)$ as a model for $\phi$ . We say that $\phi$ is free if f is free, or equivalently, if $\phi$ is a covering isometry.

We similarly define harmonic Galois covers of metric graphs.

It is clear that a harmonic $\mathfrak{G}$ -cover $\phi\;:\;\Gamma'\to\Gamma$ of metric graphs admits a model $f\;:\;G'\to G$ that is a harmonic $\mathfrak{G}$ -cover of finite graphs (the models G ^′ and G need to be sufficiently fine to avoid edge-flipping). For any $p'\in \Gamma'$ , the degree $d_\phi(p')$ is equal to the order of the stabiliser group $\mathfrak{G}_{p'}$ .

1·4. Weighted graphs, tropical curves and ramification

Graphs and metric graphs that arise as tropicalisations of algebraic curves come equipped with an additional vertex weight function that records local genera. These weights allow us to capture the auxiliary phenomenon of ramification for harmonic morphisms. We recall the definitions.

A weighted graph is a pair (G,g), where G is a finite graph and $g\;:\;V(G)\to \mathbb{Z}_{\geq 0}$ is a function, where g(v) is called the genus of the vertex v. Similarly, a tropical curve $(\Gamma,g)$ is a metric graph $\Gamma$ together with a function $g\;:\;\Gamma\to \mathbb{Z}_{\geq 0}$ with finite support. When choosing a model $(G,\ell)$ for a tropical curve $(\Gamma,g)$ , we assume that each point $x\in \Gamma$ with $g(x)>0$ corresponds to a vertex, and not to an interior point of an edge or a leg, so that (G, g) is a weighted graph. A harmonic morphism of tropical curves is a harmonic map of the underlying metric graphs.

Let (G, g) be a weighted graph. We define the Euler characteristic $\chi(v)$ of a vertex $v\in V(G)$ as

\begin{align*} \chi(v)=2-2g(v)-\textrm{val}(v).\end{align*}

Now let $f\;:\;G'\to G$ be a harmonic morphism of weighted graphs (G ^′, g ^′) and (G, g). We define the ramification degree of f at a vertex $v\in V(G')$ to be the quantity

(4) \begin{equation}\textrm{Ram}_f(v')=d_f(v')\chi(f(v'))-\chi(v').\end{equation}

We say that f is unramified if it satisfies the local Riemann–Hurwitz condition $\textrm{Ram}_f(v')=0$ for all $v'\in V(G')$ , where we note that, in contrast to the algebraic setting, it is possible for the ramification degree at a vertex to be negative. A harmonic morphism $\phi\;:\;\Gamma'\to \Gamma$ of tropical curves is unramified if it has an unramified model. Our definition of ramification was introduced in [ Reference Ulirsch and ZakharovUZ19 ], and is equivalent to the standard definition found in [ Reference Amini, Baker, Brugallé and RabinoffABBR15a ] or [ Reference Cavalieri, Markwig and RanganathanCMR16 ].

3·1. Compactifying the moduli space of $\mathfrak{G}$ -covers

Let $X\rightarrow S$ be a family of smooth projective curves of genus $g\geq 2$ with n marked disjoint sections $s_1,\ldots, s_n\in X(S)$ . A $\mathfrak{G}$ -cover of X is a finite morphism $X'\rightarrow X$ together with an operation of $\mathfrak{G}$ on X ^′ over X that is a principal $\mathfrak{G}$ -bundle on the complement of the sections, as well as a marking $s'_{ij}\in X'(S)$ of the disjoint preimages of the $s_i$ , indexed by $i=1,\ldots, n$ and $j=1,\ldots, k_i$ . Denote by $\mathcal{H}_{g,\mathfrak{G}}$ the moduli space of connected $\mathfrak{G}$ -covers of smooth curves of genus g (see e.g. [ Reference Romagny and WewersRW06 ] for a construction). There is a good notion of a limit object as X degenerates to a stable curve, as introduced in [ Reference Abramovich, Corti and VistoliACV03 ].

We emphasise that the $\mathfrak{G}$ -action is part of the data; so, in particular, an isomorphism between two admissible $\mathfrak{G}$ -covers has to be a $\mathfrak{G}$ -equivariant isomorphism. As explained in [ Reference Abramovich, Corti and VistoliACV03 ], the moduli space $\overline{\mathcal{H}}_{g,\mathfrak{G}}(\mu)$ of admissible $\mathfrak{G}$ -covers of stable n-marked curves of genus g is a smooth and proper Deligne–Mumford stack over $\textrm{Spec} \mathbb{Z}[{1}/{\vert \mathfrak{G}\vert}]$ that contains the locus $\mathcal{H}_{g,\mathfrak{G}}(\mu)$ of $\mathfrak{G}$ -covers of smooth curves of ramification type $\mu$ as an open substack. The complement of $\mathcal{H}_{g,\mathfrak{G}}(\mu)$ is a normal crossing divisor.

Remark 3·2. Although closely related, the moduli space $\overline{\mathcal{H}}_{g, \mathfrak{G}}(\mu)$ is actually not quite the same as the one constructed in [ Reference Abramovich, Corti and VistoliACV03 ]. The quotient

\begin{equation*}\big[\overline{\mathcal{H}}_{g, \mathfrak{G}}(\mu)/S_{k_1}\times \cdots \times S_{k_{n}}\big]\end{equation*}

which forgets about the order of the marked sections on $s'_{ij}$ of X ^′ over S for $i=1,\ldots,n$ and $j=1,\ldots,k_i$ , is equivalent to a connected component of the moduli space of twisted stable maps to $\textbf{B}\mathfrak{G}$ in the sense of [ Reference Abramovich and VistoliAV02, Reference Abramovich, Corti and VistoliACV03 ], indexed by ramification profile and decomposition into connected components. Our variant of this moduli space $\overline{\mathcal{H}}_{g, \mathfrak{G}}(\mu)$ , with ordered sections on X ^′, has also appeared in [ Reference Schmitt and van ZelmSvZ20 ] and in [ Reference Jarvis, Kaufmann and KimuraJKK05 ] (the latter permitting admissible covers with possibly disconnected domains).

An object in $\overline{\mathcal{H}}_{g, \mathfrak{G}}(\mu)$ is technically not an admissible $\mathfrak{G}$ -cover $X'\rightarrow X$ but rather a $\mathfrak{G}$ -cover $X'\rightarrow \mathcal{X}$ of a twisted stable curve $\mathcal{X}$ . A twisted stable curve $\mathcal{X}\rightarrow S$ is a Deligne–Mumford stack $\mathcal{X}$ with sections $s_1,\ldots, s_n\colon S\rightarrow \mathcal{X}$ whose coarse moduli space $X\rightarrow S$ is a family of stable curves over S with n marked sections (also denoted by $s_1,\ldots, s_n$ ) such that:

1. (1) the smooth locus of $\mathcal{X}$ is representable by a scheme;

2. (2) the singularities are étale-locally given by $\big[\{x'y'=t\}/\mu_r\big]$ for $t\in\mathcal{O}_S$ , where $\zeta\in\mu_r$ acts by $\zeta\cdot(x',y')=(\zeta x',\zeta^{-1}y')$ . In this case the singularity in X ^′ is locally given by $xy=t^{r}$ ;

3. (3) the stack $\mathcal{X}$ is a root stack $\big[\sqrt[r_i]{s_i/X}\big]$ along the section $s_i$ for all $i=1,\ldots, n$ ;

The two notions are naturally equivalent: given an admissible $\mathfrak{G}$ -cover $X'\rightarrow X$ , the associated twisted $\mathfrak{G}$ -cover is given by $X'\rightarrow [X'/\mathfrak{G}]$ . Conversely, given a twisted $\mathfrak{G}$ -cover $X'\rightarrow \mathcal{X}$ in the corresponding connected component, the composition $X'\rightarrow\mathcal{X}\rightarrow X$ with the morphism to the coarse moduli space X is an admissible $\mathfrak{G}$ -cover. We refer the interested reader to [ Reference Bertin and RomagnyBR11 ] for an alternative construction.

3·2. From algebraic to tropical $\mathfrak{G}$ -covers

We now explain how to construct unramified harmonic $\mathfrak{G}$ -covers of weighted graphs and tropical curves from algebraic $\mathfrak{G}$ -covers.

The operation of $\mathfrak{G}$ on $X'_{0}$ induces an operation of $\mathfrak{G}$ on G ^′ for which the map $f\;:\;G'\rightarrow G$ is $\mathfrak{G}$ -invariant. By Definition 3·1 (iii) and (iv), the stabiliser of every edge $e'_i$ and of every leg $l'_{ij}$ is a cyclic group of order $r_i$ and $r_{ij}$ , respectively. Since $F_0\;:\;X'_{0}\rightarrow X_0$ is a principal $\mathfrak{G}$ -bundle away from the nodes, the operation of $\mathfrak{G}$ on the fiber over each point in $X_0$ is transitive and so $f\;:\;G'\rightarrow G$ is a harmonic $\mathfrak{G}$ -cover. Applying the Riemann–Hurwitz formula to the restriction of $F_0$ to each irreducible component of $X'_{0}$ , we observe that f is unramified.

We note that the models $G_{\mathcal{X}'}$ and $G_{\mathcal{X}}$ depend on the choice of extension $\mathcal{F}$ , but the tropical curves $\Gamma_{X'}$ and $\Gamma_X$ do not.

The map $\phi\colon \Gamma_{X'}\rightarrow \Gamma_{X}$ may also seen to be harmonic by [ Reference Amini, Baker, Brugallé and RabinoffABBR15a , theorem A] upon identifying $\Gamma_{X'}$ and $\Gamma_X$ with the non-Archimedean skeletons of $(X')^{\textrm{an}}$ and $X^{\textrm{an}}$ , respectively. The morphism $\phi\;:\;\Gamma_{X'}\to \Gamma_X$ is unramified because f is unramified.

3·3. A modular perspective on tropicalisation

Following the recipe in [ Reference Cavalieri, Markwig and RanganathanCMR16 , section 3·2·3] one may construct a tropical moduli space $\mathcal{H}^{{\textrm{trop}}}_{g,\mathfrak{G}}(\mu)$ as a generalised cone complex that parametrises isomorphism classes of unramified harmonic $\mathfrak{G}$ -covers with dilation type $\mu$ along the marked legs.

Let us now work over an algebraically closed non-Archimedean field K, whose residue characteristic is either zero or coprime to $\vert \mathfrak{G}\vert$ . Denote by $\mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)$ the Berkovich analytic spaceFootnote ¹ associated to $\mathcal{H}_{g,\mathfrak{G}}(\mu)$ . The process described in Section 3·2 above defines a natural tropicalisation map

\begin{equation*}\begin{split}{\textrm{trop}}_{g,\mathfrak{G}}(\mu)\colon \mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)&\longrightarrow H_{g,\mathfrak{G}}^{{\textrm{trop}}}(\mu)\\\big[X'\rightarrow X,s_i,s'_{ij}\big]& \longmapsto \big[(\Gamma_{X'},g')\rightarrow(\Gamma_X,g)\big]\end{split}\end{equation*}

that associates to an admissible $\mathfrak{G}$ -cover $X'\rightarrow X$ of smooth curves over a non-Archimedean extension L of K an unramified tropical $\mathfrak{G}$ -cover $\Gamma_{X'}\rightarrow \Gamma_X$ of the dual tropical curve $\Gamma_X$ of X.

Since the boundary of $\overline{\mathcal{H}}_{g,\mathfrak{G}}(\mu)$ has normal crossings, the open immersion $\mathcal{H}_{g,\mathfrak{G}}(\mu)\hookrightarrow \overline{\mathcal{H}}_{g,\mathfrak{G}}(\mu)$ is a toroidal embedding in the sense of [ Reference Kempf, Faye Knudsen, Mumford and Saint-DonatKKMSD73 ]. Therefore, as explained in [ Reference Abramovich, Caporaso and PayneACP15, Reference ThuillierThu07, Reference UlirschUli21 ], there is a natural strong deformation retraction $\rho_{g,\mathfrak{G}}\colon \mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)\rightarrow \mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)$ onto a closed subset of $\mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)$ that carries the structure of a generalised cone complex, the non-Archimedean skeleton $\Sigma_{g,\mathfrak{G}}(\mu)$ of $\mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)$ . Expanding on [ Reference Cavalieri, Markwig and RanganathanCMR16 , theorem 1 and 4], we have:

Theorem 3·5. The tropicalisation map ${\textit{trop}}_{g,\mathfrak{G}}(\mu)\colon\mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)\longrightarrow H_{g,\mathfrak{G}}^{{\textit{trop}}}(\mu)$ factors through the retraction to the non-Archimedean skeleton $\Sigma_{g,\mathfrak{G}}(\mu)$ of $\mathcal{H}_{g,\mathfrak{G}}^{\textrm{an}}(\mu)$ , so that the restriction

\begin{equation*}{\textit{trop}}_{g,\mathfrak{G}}(\mu)\colon \Sigma_{g,\mathfrak{G}}(\mu)\longrightarrow H^{{\textit{trop}}}_{g,\mathfrak{G}}(\mu)\end{equation*}

to the skeleton is a finite strict morphism of generalised cone complexes. Moreover, the diagram

(6)

commutes.

In other words, the restriction of ${\textrm{trop}}_{g,\mathfrak{G}}(\mu)$ onto a cone in $\Sigma_{g,\mathfrak{G}}(\mu)$ is an isomorphism onto a cone in $H^{{\textrm{trop}}}_{g,\mathfrak{G}}(\mu)$ and every cone in $H_{g,\mathfrak{G}}^{{\textrm{trop}}}(\mu)$ has at most finitely many preimages in $\Sigma_{g,\mathfrak{G}}(\mu)$ . Theorem 3·5, in particular, implies that the tropicalisation map ${\textrm{trop}}_{g,\mathfrak{G}}(\mu)$ is well-defined, continuous, and proper.

The proof is almost word for word the same as the one of [ Reference Cavalieri, Markwig and RanganathanCMR16 , theorems 1 and 4]. We need to observe that the construction in [ Reference Cavalieri, Markwig and RanganathanCMR16 ] is compatible with the $\mathfrak{G}$ -operation on both the algebraic and the tropical side. Moreover, using [ Reference UlirschUli21 , section 4·5], one can extend the construction of a non-Archimedean skeleton from [ Reference Abramovich, Caporaso and PayneACP15, Reference ThuillierThu07 ] to a possibly non-trivially valued base field K. We leave the details to the avid reader, since the statement of Theorem 3·5 is not strictly used in the remainder of this paper.

4·1. From Galois covers to extended homology

Let K be a non-Archimedean field with valuation ring R and residue field k, whose characteristic p is either zero or coprime to $\vert \mathfrak{A}\vert$ . Let $F\colon X'\to X$ be a finite $\mathfrak{A}$ -cover of smooth projective curves over K (where X ^′ may be disconnected), which is ramified precisely at n ^′ marked ramification points $p'_{1},\ldots, p'_{n'}\in X'$ over a collection of marked branch points $p_1,\ldots, p_n\in X$ . Let $\mathcal{F}\colon\mathcal{X}'\to \mathcal{X}$ be an extension of $X'\to X$ to a family of admissible $\mathfrak{A}$ -covers over R (where we may have to replace K by a finite extension, as above). Let $\phi\colon \Gamma_{X'}\rightarrow \Gamma_{X}$ be the induced tropical harmonic $\mathfrak{A}$ -cover with model $f\;:\;G_{\mathcal{X}'}\to G_{\mathcal{X}}$ (which depends on the choice of $\mathcal{F}$ extending F).

Let $v\in V(G_{\mathcal{X}})$ be a vertex, then the smooth locus $X^*_v$ of the irreducible component $X_v$ is a genus g(v) curve over k with $\textrm{val}(v)$ punctures. The $\mathfrak{A}$ -cover $\mathcal{F}^{-1}(X^*_v)\to X^*_v$ is determined by a monodromy representation $m_v\;:\;\pi_1^{{\acute{e}t}}(X^*_v,x_0)\to \mathfrak{A}$ . Since $\mathfrak{A}$ is abelian, the choice of base point is irrelevant, and the representation can be recorded by a tuple of elements of $\mathfrak{A}$ in the following way. Let

\begin{align*}& \Pi_{g(v),\textrm{val}(v)}\\[5pt] &=\big\langle \alpha_1,\ldots,\alpha_{g(v)},\beta_1\ldots,\beta_{g(v)},\gamma_1,\ldots,\gamma_{\textrm{val}(v)}\ \mid \ [\alpha_1,\beta_1]\cdots[\alpha_{g(v)},\beta_{g(v)}]\gamma_1\cdots\gamma_{\textrm{val}(v)}=1\big\rangle\end{align*}

be the fundamental group of a genus g(v) Riemann surface with $\textrm{val}(v)$ punctures, where the $\gamma_j$ are small loops around the punctures. By a theorem of Grothendieck (see e.g. [ Reference SzamuelySza09 , theorem 4·9·1]), the étale fundamental group $\pi_1^{{\acute{e}t}}(X^*_v,x_0)$ is the profinite completion of $\Pi_{g(v),\textrm{val}(v)}$ when $p=0$ and the prime-to-p profinite completion of $\Pi_{g(v),\textrm{val}(v)}$ when $p>0$ . Since $\vert \mathfrak{A}\vert$ is coprime to p, every continuous homomorphism $\pi_1^{{\acute{e}t}}(X^*_v,x_0)\rightarrow \mathfrak{A}$ (where $\mathfrak{A}$ is equipped with the discrete topology) is uniquely determined by a homomorphism $\varphi\;:\;\Pi_{g(v),\textrm{val}(v)}\rightarrow \mathfrak{A}$ that factors as

\begin{equation*}\Pi_{g(v),\textrm{val}(v)}\longrightarrow \pi_1^{{\acute{e}t}}(X^*_v,x_0) \longrightarrow \mathfrak{A} \ .\end{equation*}

Hence the monodromy representation $m_v\colon \pi_1^{{\acute{e}t}}(X^*_v,x_0)\to \mathfrak{A}$ is uniquely determined by the images

\begin{align*} \xi(v)_i=\varphi(\alpha_i)\in \mathfrak{A}\quad \textrm{and} \quad \xi(v)_{g(v)+i}=\varphi(\beta_i)\in\mathfrak{A}\quad \textrm{for}\quad i=1,\ldots,g(v),\end{align*}

of the $\alpha_i$ and $\beta_i$ , which may be arbitrary, as well as the images

\begin{align*} \eta(h)=\varphi(\gamma_j)\in \mathfrak{A}\quad\textrm{for}\quad j=1,\ldots,\textrm{val} (v),\end{align*}

where $h\in H(G_{\mathcal{X}})$ is the half-edge corresponding to the jth puncture on $X_v$ . We note that $\eta(h)$ acts by multiplication by a primitive rth root of unity in an étale neighbourhood of $p_{h'}$ , where $r=d_{\phi}(h')$ . The unique relation in the group $\Pi_{g(v),\textrm{val}(v)}$ implies that the elements $\eta(h)$ satisfy

\begin{align*} \sum_{j=1}^{\textrm{val}(v)}\varphi(\gamma_j)=\sum_{h\in T_vG_{\mathcal{X}}}\eta(h)=0.\end{align*}

Furthermore, for each pair of nodes $e=\{h,h'\}$ we have $\eta(h)+\eta(h')=0$ .

In other words, to an algebraic $\mathfrak{A}$ -cover $X'\to X$ we associate the following data on the graph $G_{\mathcal{X}}$ :

1. (1) an element $\eta(h)\in \mathfrak{A}$ for each $h\in H(G_{\mathcal{X}})$ , so that $\eta(h)+\eta(h')=0$ for any edge $e=\{h,h'\}$ in $E(G_\mathcal{X})$ and $\sum_{h\in T_v G_\mathcal{X}} \eta(h)=0$ for any vertex $v\in V(G_\mathcal{X})$ ;

2. (2) an element $\xi(v)\in \mathfrak{A}^{2g(v)}$ for every vertex $v\in V(G_\mathcal{X})$ .

The collection of all $\eta(h)$ is nothing but a class $\eta\in H_1(G_\mathcal{X},\mathfrak{A})$ in the simplicial homology of the graph $G_{\mathcal{X}}$ with coefficients in $\mathfrak{A}$ . Similarly, each $\mathfrak{A}^{2g(v)}$ can be thought of as the simplicial homology group (with coefficients in $\mathfrak{A}$ ) of an infinitesimal genus g(v) graph located at the vertex v. This motivates the following definition.

Definition 4·2. Let (G,g) be a weighted graph. The extended homology group of G with coefficients in $\mathfrak{A}$ is the finite abelian group

\begin{align*} H^{\textrm{ext}}_1(G,\mathfrak{A})=H_1(G,\mathfrak{A})\oplus \bigoplus_{v\in V(G)} \mathfrak{A}^{2g(v)}.\end{align*}

Given a harmonic $\mathfrak{A}$ -cover $f\;:\;G_{\mathcal{X}'}\to G_{\mathcal{X}}$ of weighted graphs that is the tropicalisation of an algebraic $\mathfrak{A}$ -cover $F\;:\;X'\to X$ , the datum $(\eta,\xi)\in H_1^{\textrm{ext}}(G_{\mathcal{X}},\mathfrak{A})$ defined above is called the $\mathfrak{A}$ -monodromy datum associated to the cover.

We note that if $|\mathfrak{A}|$ is coprime to p, then the group of continuous homomorphisms from $\pi_1^{{\acute{e}t}}(X,x)$ to $\mathfrak{A}$ is naturally identified with $H^1_{{\acute{e}t}}(X,\mathfrak{A})$ (see [ Reference MilneMil13 , example 11·3]), and hence the discussion above provides us with a natural homomorphism

\begin{equation*}H^1_{{\acute{e}t}}(X,\mathfrak{A})\longrightarrow H^{\textrm{ext}}_1(G_{\mathcal{X}},\mathfrak{A}).\end{equation*}

Note that we pass from cohomology to homology, as $G_{\mathcal{X}}$ is the dual graph of the special fiber $\mathcal{X}_0$ .

4·2. Extended homology, dilation and realisability

We now observe that the $\mathfrak{A}$ -dilation datum of a harmonic $\mathfrak{A}$ -cover $f\;:\;G_{\mathcal{X}'}\to G_{\mathcal{X}}$ obtained by tropicalising an algebraic $\mathfrak{A}$ -cover can be read off from the $\mathfrak{A}$ -monodromy datum $(\eta,\xi)$ . Indeed, let $v'\in V(G_{\mathcal{X}'})$ be a vertex mapping to $v=f(v')$ . The restricted $\mathfrak{A}$ -cover $f^{-1}(X^*_v)\to X^*_v$ corresponds to the monodromy representation $m_v\;:\;\pi_1^{{\acute{e}t}}(X_v^*,x_0)\to \mathfrak{A}$ , so an element of $\mathfrak{A}$ preserves the irreducible component $X^*_{v'}\subseteq f^{-1}(X^*_v)$ (in other words, fixes v ^′) if and only if it is in the image of $m_v$ , which is generated by the images of the $\alpha_i$ , $\beta_i$ , and $\gamma_j$ . Similarly, the stabiliser of a node $p_{h'}\in \mathcal{X}'_0$ mapping to $p_h\in \mathcal{X}_0$ is generated by $\eta(h)\in \mathfrak{A}$ . Hence we give the following definition.

We are now ready to state and prove our realisability criterion.

Conversely, suppose that the $\mathfrak{A}$ -dilation datum D of an unramified harmonic $\mathfrak{A}$ -cover $f\;:\;G'\to G$ is associated to an element $(\eta,\xi)\in H_1^{\textrm{ext}}(G,\mathfrak{A})$ . We reverse the procedure and construct an admissible $\mathfrak{A}$ -cover $F_0\;:\;X'_{0}\to X_0$ over k tropicalizing to f, as follows. For each vertex $v\in V(G)$ , choose a smooth k-curve $X^*_v$ of genus g(v) with $|T_vG|$ punctures. The monodromy element $(\eta,\xi)$ induces a monodromy representation $m_v\;:\;\pi_1^{{\acute{e}t}}(X^*_v,x_0)\to \mathfrak{A}$ at each $v\in V(G)$ and hence an $\mathfrak{A}$ -cover of each $X^*_v$ . We then glue these covers according to the incidence data of the graphs G’ and G to obtain an admissible $\mathfrak{A}$ -cover $F_0\;:\;X'_{0}\to X_0$ , where the $X_v$ are the irreducible components of $X_0$ . Hence f is realisable.

If $f\;:\;G'\to G$ is the underlying model for a harmonic $\mathfrak{A}$ -cover $\phi\;:\;\Gamma'\to \Gamma$ , then the admissible $\mathfrak{A}$ -cover $F_0\;:\;X'_{0}\to X_0$ can be smoothened to a family $\mathcal{F}\;:\;\mathcal{X}'\to \mathcal{X}$ by the smoothness of the moduli space of admissible $\mathfrak{A}$ -covers over $\textrm{Spec} \mathbb{Z}[{1}/{\vert \mathfrak{A}\vert}]$ (see [ Reference Abramovich, Corti and VistoliACV03 , theorem 3·0·2]). Alternatively, one may also use the smoothening result for harmonic covers of metrised curve complexes from [ Reference Amini, Baker, Brugallé and RabinoffABBR15a ] and observe that their smoothing is compatible with group operations.

This criterion can be used to establish elementary graph-theoretic restrictions on realisable harmonic $\mathfrak{A}$ -covers. First, we note that if the dilation group of any half-edge is not cyclic, then the cover is not realisable. Now let G be a weighted graph with a bridge edge $e\in E(G)$ . Any $\mathfrak{A}$ -monodromy datum $(\eta,\xi)\in H_1^{\textrm{ext}}(G,\mathfrak{A})$ vanishes on e, so the dilation group of any realisable $\mathfrak{A}$ -cover is trivial along e. Hence we have established the following necessary condition for realisability.

In the next section, we show that, for most simple abelian groups, this condition is in fact sufficient. Similarly, if G has a pair of parallel edges $e_1$ and $e_2$ , then the dilation groups of any realisable $\mathfrak{A}$ -cover are equal on $e_1$ and $e_2$ , since $\eta(e_2)=\pm \eta(e_1)$ for any $(\eta,\xi)\in H_1^{\textrm{ext}}(G,\mathfrak{A})$ .