Moduli spaces of compact RCD(0,N)-structures

The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$\end{document}RCD(0,N)-structures. First, we relate the convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$\end{document}RCD(0,N)-structures on a space to the associated lifts’ equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$\end{document}RCD(0,N)-structures that have non-trivial rational homotopy groups.


Introduction
One of Riemannian geometry's most fundamental problems is studying metrics satisfying a particular curvature constraint on a fixed smooth manifold.Three thoroughly studied types of curvature are the sectional, the Ricci, and the scalar curvature; common curvature constraints are lower (resp.upper) bounds on the corresponding curvature.Also, when a smooth manifold admits a metric of the desired type, it is interesting to describe such metrics' space.A way to tackle this problem is to study the topological properties of the associated moduli space, i.e., the quotient of the space of all metrics satisfying the curvature condition by isometry equivalence.In the last decades, moduli spaces of metrics with positive scalar curvature (resp.negative sectional curvature) have been studied intensively (see [38] for a comprehensive introduction).Yet, there are not as many results on moduli spaces of non-negatively Ricci curved metrics.In 2017, Tuschmann and Wiemeler published the first result on these moduli spaces' homotopy groups (see Theorem 1.1 in [37]).
Until recently, most of the results on moduli spaces in metric geometry focused on smooth metrics.Nevertheless, Belegradek [7] recently tackled the case of nonnegatively curved Alexandrov spaces, studying the moduli space of non-negatively curved length metrics on the 2-sphere.Whereas Alexandrov introduced curvature lower bounds in the setting of length spaces (generalizing sectional curvature lower bounds), RCD spaces generalize lower bounds on the Ricci curvature to the setting of metric measure spaces.Indeed, roughly, RCD(0, N ) spaces should be thought as possibly non smooth spaces with dimension bounded above by N ∈ [1, ∞) and nonnegative Ricci curvature, in a synthetic sense.The first goal of the present paper is to set the foundations for studying moduli spaces of RCD(0, N )-structures.The question we will be studying in this paper is the following: , and let X be a compact topological space that admits an RCD(0, N )-structure.What can be said about the topology of the moduli space of RCD(0, N )-structures on X ?
We will start by recalling in Sect.1.1 what an RCD(0, N ) space is.Then, in Sect.1.2, we will introduce RCD(0, N )-structures on a fixed topological space, together with the associated moduli space.In Sect.1.3, we will introduce the notions of lift and push-forward of an RCD(0, N )-structure.Afterwards, in Sect.1.4, we will present concisely the Albanese map and the soul map associated to a compact topological space that admits an RCD(0, N )-structure.Finally, in Sect.1.5, we will present our main results.

RCD(0,N)-spaces
The story of RCD spaces has its roots in Gromov's precompactness Theorem (see Corollary 11.1.13 of [30]).The result states that sequences of compact Riemannian manifolds with a lower bound on the Ricci curvature, and an upper bound on both the dimension and the diameter are precompact in the Gromov-Hausdorff topology (GH topology for short).Since then, there has been much work to understand the properties of limits of such sequences, called Ricci limit spaces.In the early '00s, Cheeger and Colding published in [14,15] and [16] an extensive study of the aforementioned spaces.One important observation (already noticed by Fukaya) is that, to retain good stability properties at the limit space, it is fundamental to keep track of the Riemannian measures' behaviour associated to the approximating sequence.Since then, it is common to use the measured Gromov-Hausdorff topology (mGH topology for short), endowing Riemannian manifolds with their normalized volume measure.
A related (but slightly different) approach is to introduce a new definition of Ricci curvature lower bounds and dimension upper bound, at the more general level of possibly non-smooth metric measure spaces, which generalizes the classical notions and is stable when passing to the limit in the mGH topology.The definition of RCD spaces is an example of such a definition.Therefore, any result proven for RCD spaces (with the tools of metric measure theory) would hold a fortiori for Ricci limit spaces.
Throughout the paper, we will use the following definition of metric measure spaces.
Definition 1.1 Let (X , d, m) be a triple where (X , d) is a metric space and m is a measure on X .We say that (X , d, m) is a metric measure space (m.m.s. for short) when (X , d) is a complete separable metric space and m is a non-negative boundedly finite Radon measure on (X , d).
CD-spaces were introduced independently by Lott and Villani in [25], and by Sturm in [35] and [36].For simplicity, we only give the definition of CD(0, N )-spaces (following Definition 1.3 in [36]).An extensive study of CD-spaces is given in [39,Chapters 29 and 30].
Denote by P 2 (X , d, m) the space of probability measures that are absolutely continuous w.r.t.m and with finite variance, and let W 2 be the quadratic Kantorovitch-Wasserstein transportation distance.Let also S N (μ | m) be the Renyi entropy of μ ∈ P 2 (X , d, m) with respect to m, i.e.
The class of CD spaces includes (some) non-Riemannian Finsler structures which, from the work of Cheeger-Colding [14][15][16], cannot appear as Ricci limits.In order to single out the "Riemannian" CD structures, it is convenient to add the assumption that the Sobolev space W 1,2 is a Hilbert space [3,22] (see also [2]): indeed for a Finsler manifold, W 1,2 is a Banach space and W 1,2 is a Hilbert space if and only if the Finsler structure is actually Riemannian.Such a condition is known as "infinitesimal Hilbertianity".Definition 1.3 Given N ∈ [1, ∞), a m.m.s.(X , d, m) is an RCD(0, N ) space if it is an infinitesimally Hilbertian CD(0, N )-space.
An equivalent way to characterise the RCD(0, N ) condition is via the validity of the Bochner inequality [4,5,19] (see also [12] for the globalization of general Ricci lower bounds RCD(K , N )).

Moduli spaces of RCD(0, N)-structures on compact topological spaces
In this section, we will start by defining RCD(0, N )-structures on a fixed topological space; then, we will introduce the associated moduli space, together with its topology.Definition 1. 4 Given a topological space X and N ∈ [1, ∞), an RCD(0, N )-structure on X is a metric measure structure (X , d, m) such that: d metrizes the topology of X , Spt(m) = X , and (X , d, m) is an RCD(0, N ) space.
It is common to identify m.m.s. that are isomorphic.There are two distinct notions of isomorphisms for m.m.s.(see the discussion in Chapter 27, Section "adding the measure" in [39]).However, both notions coincide when restricted to the set of RCD(0, N )-structures on a topological space (since we imposed measures to have full support in that case).We will adopt the following definition of isomorphism between metric measure spaces.Definition 1.5 Two m.m.s.(X 1 , d 1 , m 1 ) and (X 2 , d 2 , m 2 ) are isomorphic when there is a bijective isometry φ : (X 1 , d 1 ) → (X 2 , d 2 ) such that φ * m 1 = m 2 .

Theorem 1.1 The compact Gromov-Hausdorff-Prokhorov distance d c
GHP (see [1,Section 2.2]) is a complete separable metric on the set X of isomorphism classes of compact metric measure spaces.Moreover, d c GHP metrizes the mGH topology (see for instance [39,Definition 27.30]).
We are now in position to introduce the moduli space M 0,N (X ) of RCD(0, N )structures on a compact topological space X ; this will be the main object of study in the paper.Notation 1.1 Let N ∈ [1, ∞), and let X be a compact topological space that admits an RCD(0, N )-structure.We introduce the following spaces: (i) RCD(0, N ) ⊂ X is the set of isomorphism classes of compact RCD(0, N ) spaces with full support, endowed with the mGH-topology (seen as a subspace of X), (ii) R 0,N (X ) is the set of all RCD(0, N )-structures on X , (iii) M 0,N (X ) is the quotient of R 0,N (X ) by isomorphisms, endowed with the mGHtopology (seen as a subspace of RCD(0, N )).
We call M 0,N (X ) the moduli space of RCD(0, N )-structures on X .

Lift and push-forward
To our aims, a fundamental result is the existence of a universal cover for an RCD space (see [27,Theorem 1.1]).
, and let X be a compact topological space that admits an RCD(0, N )-structure.Then X admits a universal cover p : X → X .We denote π 1 (X ) the associated group of deck transformations, also called the revised fundamental group of X.

Remark 1.1
In Theorem 1.2, the universal cover must be understood in a general sense (as defined in [34, Chap.2, Sect.5]).To be precise, it is not known whether it is simply connected or not.In particular, the revised fundamental group π 1 (X ) is a quotient of the fundamental group π 1 (X ) of X , and may a priori not be isomorphic to π 1 (X ).
As in the case of a Riemannian manifold, it is possible to lift an RCD(0, N )structure on a compact topological space to its universal cover, and, conversely, to push-forward an equivariant RCD(0, N )-structure on the universal cover back to the base space.Indeed, let us fix a real number N ∈ [1, ∞), a compact topological space X that admits an RCD(0, N )-structure, and denote p : X → X the universal cover of X .We will see in Sect.2.1 that: ) is a local isomorphism, and π 1 (X ) acts by isomorphism on ( X , d, m) (see Corollary 2.1); • given an RCD(0, N )-structure ( X , d, m) on X such that π 1 (X ) acts by isomorphism on ( X , d, m), there exists a unique RCD(0, N )-structure (X , d, m) on X (called the push-forward of ( X , d, m)) such that p : Theorem A, which will be introduced in Sect.1.5, relates the convergence of RCD(0, N )-structures on X to the convergence of the associated lifts.

Soul map and Albanese map
In this section, we fix a real number N ∈ [1, ∞), a compact topological space X that admits an RCD(0, N )-structure, and denote p : X → X the universal cover of X .A special property enjoyed by RCD(0, N ) spaces is the existence of splittings.More precisely, given an RCD(0, N )-structure (X , d, m) on X , and denoting ( X , d, m) the associated lift (see Sect. 1.3), there exists (thanks to Theorem 1.3 in [27]) an isomorphism: where k ∈ N ∩ [0, N ] is called the degree of φ, R k is endowed with the Euclidean distance and the Lebesgue measure, and (X , d, m) is a compact RCD(0, N − k)-space such that π 1 (X ) = 0; the space X is called the soul of φ (see Theorem 1.2 for the definition of π 1 ).Such a map φ is called a splitting of ( X , d, m), and induces an isomorphism: Moreover, the revised fundamental group π 1 (X ) acts by isomorphism onto ( X , d, m).Therefore, applying φ * and projecting onto Iso(R k ), we get a group homomorphism ρ φ R : π 1 (X ) → Iso(R k ).In Sect.2.2, we will prove the following properties: • the degree k does not depend either on the chosen splitting φ, or the chosen ) is a crystallographic subgroup of Iso(R k ); moreover, up to conjugating with an affine transformation, φ does not depend either on the chosen splitting φ, or the chosen RCD(0, N )-structure (X , d, m) on X (see Proposition 2.7).Therefore, it is possible to introduce k(X ) := k (called the splitting degree of X ), and the set (X ) of crystallographic subgroups of Iso(R k ) that are conjugated to φ by an affine transformation (called the crystallographic class of X ), both being topological invariants of X .
Thanks to the above discussion, to any RCD(0, N )-structure (X , d, m) on X with lift ( X , d, m), and to any splitting φ of ( X , d, m), we can associate: with orbifold fundamental group equal to φ ∈ (X ) (see the discussion preceding Definition 2.3).We will denote A(X ) the set of compact flat k(X )-dimensional orbifolds whose orbifold fundamental group belong to (X ) (called the Albanese class of X ).

Remark 1.2
Let (M N , g) be a compact N -dimensional Riemannian manifold with non-negative Ricci curvature, and such that π 1 (M) = Z k .Observe that, in that case, (M, d g , m g ) is an RCD(0, N ) space, where d g is the geodesic distance and m g is the Riemannian measure.It is possible to show that the orbifold obtained following the discussion above is nothing but the usual Albanese variety of (M, g) (up to isometry).
In Sect.2.4.2, we will see that, up to isomorphism, the orbifold (resp.the soul) does not depend either on the choice of the splitting map φ, or on the isomorphism class of (X , d, m) (see Lemma 2.1).Therefore, we will be able to introduce: • the Albanese map A : M 0,N (X ) → M flat (A(X )) associated to X , where M flat (A(X )) is the quotient of A(X ) by isometry equivalence (endowed with the GH topology); • the soul map S : M 0,N (X ) → RCD(0, N − k(X )) associated to X ; such that for any RCD(0, N )-structure (X , d, m) on X with lift ( X , d, m) and any splitting φ of ( X , d, m) with soul (X , d, m), we have m] (where brackets denote the equivalence class in the appropriate moduli space).
Theorem B, which will be presented in Sect.1.5, states the continuity of the Albanese and soul maps.

Main results
Our first result relates the convergence of RCD(0, N )-structures on a compact topological space to the convergence of the associated lifts.), let X be a compact topological space that admits an RCD(0, N )-structure, and denote p : X → X the universal cover of X .Assume that for every n ∈ N ∪ {∞}: • ∞ , * ∞ ) in the equivariant pmGH topology.As we will observe at the end of Sect.2.4.1,Theorem A implies the following corollary, which is particularly useful when computing the homeomorphism type of specific examples of moduli spaces (see for example the case of RP 2 in [28]).
Corollary A Let N ∈ [1, ∞), let X be a compact topological space that admits an RCD(0, N )-structure, and denote p : X → X the universal cover of X .
Then the lift map: is a homeomorphism (introduced in Sect.2.4.1),where M p,eq 0,N ( X ) and M p 0,N (X ) are respectively the moduli space of equivariant pointed RCD(0, N )-structures on X and the moduli space of pointed RCD(0, N )-structures on X (introduced in Sect. 2

.3).
Observe that it is more straightforward to obtain Corollary A by using Theorem A than its equivalent version given in Remark 1.3.
Our next result states the continuity of the Albanese map and the soul map associated to a compact topological space that admits an RCD(0, N )-structure (with N ∈ [1, ∞)).On the first hand, this result is essential when computing the homeomorphism type of specific examples of moduli spaces (see for example the case of the Möbius band M 2 and the finite cylinder S 1 × [0, 1] in [28]).On the other hand, the continuity of the Albanese map will be crucial in the proof of Theorem C.
, and let X be a compact topological space that admits an RCD(0, N )-structure.Then, the Albanese map A : M 0,N (X ) → M flat (A(X )) and the soul map S : M 0,N (X ) → RCD(0, N − k(X )) are continuous, where M flat (A(X )) and RCD(0, N − k(X )) are respectively endowed with the GH and mGH topology.
Let us recall that if X is a compact topological space that admits an RCD(0, 2)structure, then the moduli space M 0,2 (X ) is contractible (see Theorem 1.1 in [28]).Theorem C should be put in contrast with that result since it shows that the topology of moduli spaces of RCD(0, N )-structures is not always as trivial.Moreover, Theorem C can also be seen as a non-smooth analogue of Theorem 1.1 in [37].
) and let X be a compact topological space that admits an RCD(0, N )-structure such that π 1 (X ) = 0 (see Theorem 1.2 for the definition of π 1 (X )).In addition, let Y be either S 1 ×K 2 (where K 2 is the Klein bottle) or a torus of dimension k ≥ 4 such that k = 8, 9, 10.Then, the moduli space M 0,N +dim(Y ) (X × Y ) has non-trivial higher rational homotopy groups.
Thanks to Theorem C, we immediately obtain the following corollary, which can be seen as a non-smooth analogue of Corollary 1.2 in [37].
Corollary B For every N ≥ 3 (resp.N ≥ 4 / N ≥ 5) there exists a compact topological space X such that M 0,N (X ) is not simply connected (resp.has non-trivial third rational homotopy group / non-trivial fifth rational homotopy group).
In Sect.2, we will introduce in full details the main objects and constructions of the paper.In Sect.3, we will prove the main results.

Preliminaries
Throughout this section: • X is a compact topological space that admits an RCD(0, N )-structure, • p : X → X denotes the universal cover of X (whose existence is given by Theorem 1.2).
In Sect.2.1, we will introduce the notions of lift (resp.push-forward) of an RCD(0, N )-structure on X (resp.on X ).
In Sect.2.2, we will present splittings and use them to construct topological invariants associated to X (splitting degree, crystallographic class and Albanese class).
In Sect.2.3, we will define the moduli space of pointed RCD(0, N )-structures on X ; then, we will introduce the moduli space of equivariant pointed RCD(0, N )-structures on the universal cover X .
In Sect.2.4, we will define the lift and push-forward map (which are important to get Corollary A), and the Albanese and soul maps.

Covering space theory of RCD(0,N)-spaces
In this section, we will start by introducing δ-covers associated to an RCD(0, N )structure on X .Then, we will explain how to lift an RCD(0, N )-structure on X to the associated δ-cover.Afterwards, we will explain how the universal cover of X is related to δ-covers.Subsequently, we will explain how to lift an RCD(0, N )-structure on X to its universal cover X , and, conversely, how to push-forward an equivariant RCD(0, N )-structure on X onto X .Finally, we will introduce the Dirichlet domain associated to an RCD(0, N )-structure on X .
Before introducing δ-covers, we recall the following result (Chapter 2, Sections 4 and 5 of [34]) that associates a regular covering p U : X U → X to any open cover U of X .

Proposition 2.1
Given an open cover U of X , there exists a unique regular covering p U : X U → X (up to equivalence) such that: where π 1 (U, p U (y)) is composed of homotopy classes of loops of the form ω −1 * α * ω, where α is a loop contained in some U ∈ U and ω is a path from p U (y) to α(0).Moreover, every connected open set U ∈ U is evenly covered bu p U .
The notion of δ-cover was introduced first by Sormani and Wei to prove the existence of a universal cover for Ricci limit spaces (see Theorem 1.1 in [33]).Later, it has also been used by Mondino and Wei in [27] to prove Theorem 1.2.These covering spaces will be very important in the proof of Theorem A. We write G(δ, d) the associated group of deck transformations.
In the following result, we introduce the lift of an RCD(0, N )-structure on X to a δ-cover.Proposition 2.2 Given δ > 0 and (X , d, m) an RCD(0, N )-structure on X , there exists a unique RCD(0 d, m) is a local isomorphism.Moreover, we have the following properties:  [31], and by definition of m δ , point (iv) and (v) are satisfied.Now, we put the universal cover of X in relation with δ-covers (see Theorem 2.7 in [27] for a proof).
Theorem 2.1 Let (X , d, m) be an RCD(0, N )-structure on X , and let δ(X , d) be the supremum of all δ > 0 such that every ball of radius δ in (X , d) is evenly covered by p. Then δ(X , d) > 0, and for every δ < δ(X , d), p and p δ d are equivalent, and every equivalence map is an isomorphism between ( X , d, m) and (X δ d , d δ , m δ ).
Thanks to Proposition 2.2 and Theorem 2.1, we can introduce the lift of an RCD(0, N )-structure on X to the universal cover X .(X , d, m) be an RCD(0, N )-structure on X .There is a unique RCD(0, N )-structure ( X , d, m) on X (called the lift of (X , d, m)) such that p : ( X , d, m) → (X , d, m) is a local isomorphism.Moreover, the revised fundamental group π 1 (X ) acts by isomorphism on ( X , d, m).

Corollary 2.1 Let
The following proposition is a sort of converse to Corollary 2.1; it introduces the push-forward of an equivariant RCD(0, N )-structure on X (cf.[27,Lemma 2.18] and [26,Lemma 2.24]).Proposition 2.3 Let ( X , d, m) be an RCD(0, N )-structure on X such that π 1 (X ) acts by isomorphisms on ( X , d, m).There is a unique RCD(0, N )-structure (X , d, m) on X (called the push-forward of ( X , d, m)) such that p : ( X , d, m) → (X , d, m) is a local isomorphism.It satisfies the following properties: (i) for every x, y ∈ X , we have d(x, y) = inf{ d( x, ỹ)}, where the infimum is taken over all x ∈ p −1 (x) and ỹ ∈ p −1 (y), (ii) for every open set U ⊂ X that is evenly covered by p, we have m(U ) = m( Ũ ), where Ũ is any open set in X such that p : Ũ → U is a homeomorphism.
Proof First of all, there is obviously at most one RCD(0, N )-structure on X such that p is a local isomorphism.Then, let us define d and m as in points (i) and (ii).Observe that since X is locally compact, and since π 1 (X ) acts by isometries, m is well defined, and the infimum in the definition of d is achieved.It is then readily checked that d is a distance on X , and that m defines a measure on X (using the fact that the Borel σ -algebra of X is generated by evenly covered open sets).
Let us now show that p is a local isomorphism.Let x ∈ X and define x := p( x).
There exists an open neighborhood Ũ of x such that p : Ũ → U := p( Ũ ) is a homeomorphism.Moreover, there exists r > 0 such that B d( x, r ) ⊂ Ũ .Let us show that, for every 0 < r ≤ r , p is a homeomorphism from B d( x, r ) onto B d (x, r ).First, notice that p is distance decreasing; in particular, we have p(B d( x, r )) ⊂ B d (x, r ).Now, let y ∈ B d (x, r ).Since the infimum in the definition of d is achieved, there exists Now, let ỹ, z ∈ B d( x, r /3).Looking for a contradiction, let us suppose that d(y, z) < d( ỹ, z), where y := p( ỹ) and z := p(z).In that case, there exists ), so we should have z = z, which is the contradiction we were looking for.Hence, p is an isometry from B d( x, r /3) onto B d (x, r /3).Moreover, by definition of m, this implies that p is an isomorphism of metric measure space from Then, it is easy to check that d metrizes the topology of X and that Spt(m) = X .To conclude, we just need to show that (X , d, m) is an RCD(0, N ) space.To this aim, first of all observe that RCD(0, N ) is equivalent to RCD * (0, N ) (by the explicit form of the distortion coefficients), which in turn is equivalent to CD e (0, N ) plus infinitesimally Hilbertianity.
Observe that, since p is a local isometry, it preserves the length of curves; therefore, (X , d) is a compact geodesic space.Moreover, since X is compact and m is boundedly finite, m is necessarily a finite measure on X .Summarising: (X , d, m) is a compact geodesic space endowed with a finite measure, and it is locally isomorphic to an RCD(0, N ) space in the sense that for every point x ∈ X there exists a closed metric ball B d (x, r ).centred at x isomorphic to a closed metric ball B d( x, r ).inside the RCD(0, N ) space X .
Notice that, by triangle inequality, if y, z ∈ B d (x, r /4).then any geodesic joining them is contained in B d (x, r ).Recall also that, given two absolutely continuous probability measures with compact support in an RCD space, there exists a unique W 2 -geodesic joining them [24,Theorem 1.1].It follows that, given two absolutely continuous probability measures with compact support contained in B d (x, r /4) (which in turm is isomorphic to B d (x, r /4) ⊂ X , and X satisfies RCD(0, N )), there exists a unique W 2 -geodesic joining them, its support is contained in B d (x, r ), and it satisfies the convexity property of the CD e (0, N ) condition.
In particular, (X , d, m) satisfies the strong CD e loc (0, N ) condition in the sense of [19] and it is locally infinitesimally Hilbertian.Then, using [19,Theorem 3.25], we obtain that (X , d, m) satisfies RCD(0, N ).(X , d, m) is an RCD(0, N )-structure on X , then the pushforward of the lift of (X , d, m) is equal to (X , d, m), thanks to Proposition 2.3 and Corollary 2.1.The same is true in the other direction; if ( X , d, m) is an RCD(0, N )structure on X such that π 1 (X ) acts by isomorphisms, then the lift of the push-forward of ( X , d, m) is equal to ( X , d, m).

Remark 2.1 Observe that if
We conclude this section with the following results that introduces the Dirichlet domain associated to an RCD(0, N )-structure on X .Proposition 2.4 Let (X , d, m) be an RCD(0, N )-structure on X and let x ∈ X.We define the Dirichlet domain with center x associated to (X , d, m) by: where φ η ( ỹ) := d( ỹ, η x) − d( ỹ, x), for ỹ ∈ X .The Dirichlet domain satisfies the following two properties: (i) for every ỹ ∈ X , there exists η ∈ π 1 (X ) such that η ỹ ∈ F( x), (ii) for every ỹ ∈ F( x), we have d( x, ỹ) = d(x, y), where x := p( x) and y := p( ỹ).
Proof We start with the proof of (i).Let ỹ ∈ X and define R := d( x, ỹ).Then, ) is a compact, discrete, non empty set; hence, it contains finitely many points.In particular, there exists η ∈ π 1 (X ) such that η x ∈ B d ( ỹ, R), and such that: Now, assume that μ Hence, for every μ ∈ π 1 (X ), we have φ μ (η Now we prove (ii).Assume that ỹ ∈ F( x).We define y := p( ỹ) and we assume that β : [0, 1] → X is a minimizing geodesic from x to y.Let β be the lift of β starting at x and let η ∈ π 1 (X ) such that β(1) = η ỹ.Looking for a contradiction, let us suppose that d(x, y) , which is the contradiction we were looking for.Thus, d(x, y) ≥ d( x, ỹ), and, since p contracts distances, we have d(x, y) = d( x, ỹ).This concludes the proof.

Splittings and topological invariants
In this section, we will introduce the notion of splitting associated to an RCD(0, N )structure on X .To any splitting φ, we will associate a degree k and a Euclidean homomorphism ρ φ R : π 1 (X ) → Iso(R k ), and we will investigate the properties of (φ) = Im(ρ φ R ).We will prove that the degree k and the affine conjugacy class of (φ) do not depend either on the chosen splitting φ, or the chosen RCD(0, N )-structure on X .This will lead us to introduce the splitting degree k(X ) and the crystallographic class (X ) of X , which are topological invariants of X .Finally, we will introduce the Albanese class A(X ) of X , which consists of orbifolds whose fundamental group belong to (X ).
First of all, let us introduce the definition of splittings.
Definition 2.2 Let (X , d, m) be an RCD(0, N )-structure on X , and denote , where R k is endowed with the Euclidean distance and Lebesgue measure, k ∈ N ∩ [0, N ] is called the degree of φ, and (X , d, m) is a compact RCD(0, N − k)-space with trivial revised fundamental group called the soul of φ.
Thanks to Theorem 1.3 in [27], which in turn built on top of the Splitting Theorem for RCD(0, N ) spaces [21], we have the following existence result.Theorem 2.2 For every RCD(0, N )-structure (X , d, m) on X , the lift ( X , d, m) admits a splitting.Moreover, for every splitting φ of ( X , d, m), the group of isomorphisms of (X , d, m) × R k splits, i.e., we have: where (X , d, m) is the soul of φ and k is the degree of φ.
Thanks to Theorem 2.2, Theorem 2.1, and Proposition 2.2, we can introduce the following notations.Notation 2.1 Let (X , d, m) be an RCD(0, N )-structure on X and let φ be a splitting of its lift ( X , d, m) with degree k and soul (X , d, m).We write: We call ρ φ the soul homomorphism associated to φ (resp.the Euclidean homomorphism associated to φ) and we write The next result shows that the kernel and the image of the Euclidean homomorphism associated to a splitting enjoy particular group structures.Proposition 2.5 Let (X , d, m) be an RCD(0, N )-structure on X and let φ be a splitting of ( X , d, m) with degree k and soul (X , d, m).Then, K (φ) is a finite normal subgroup of π 1 (X ) and (φ) is a crystallographic subgroup of Iso(R k ) (i.e. it acts cocompactly and discretly on R k ).
Now, let us show that (φ) acts cocompactly on R k .Thanks to the first isomorphism theorem for topological spaces, there is a continuous map μ such that the following diagram is commutative: where q i (i ∈ {1, 2}) are the quotient maps.Moreover, μ is surjective since is compact, being the image of a compact topological space by a continuous surjective map.In conclusion, (φ) acts cocompactly on R k .Let us prove that (φ) acts discretely on R k (i.e. its orbits are discrete subsets of R k ).First, observe that it is sufficient to prove that (φ) acts properly on R k .To prove this, let K be a compact subset of R k and let us show that there are only finitely many elements g ∈ (φ) such that gK ∩ K = ∅.By definition of (φ), we have: The following corollary of Proposition 2.5 defines the splitting degree of X (cf.[26,Proposition 2.25]).

Corollary 2.2 (Splitting degree k(X ))
The revised fundamental group π 1 (X ) is a finitely generated group which has polynomial growth of order k(X ) ∈ N ∩ [0, N ].Moreover, given any RCD(0, N )-structure (X , d, m) on X with lift ( X , d, m), the degree of any splitting φ of ( X , d, m) is equal to k(X ).We call k(X ) the splitting degree of X.
where k ∈ [0, N ] ∩ N is the degree of φ.We need to prove that π 1 (X ) has polynomial growth of order k.By Bieberbach's first Theorem (see Theorem 3.1 in [13]), (φ) admits a normal subgroup In particular, (φ) ∩ R k is finitely generated, has polynomial growth of order k, and is a normal subgroup of (φ) with finite index; thus, (φ) is also finitely generated and has polynomial growth of order k.Now, π 1 (X )/K (φ) is isomorphic to (φ); hence it is finitely generated with polynomial growth of order k.However, K (φ) is finite and is a normal subgroup of π 1 (X ); thus, π 1 (X ) is also finitely generated and has polynomial growth of order k.
The revised fundamental group satisfies the following additional group property (which will be crucial in the proof of Theorem A).

Proposition 2.6
The revised fundamental group π 1 (X ) is a Hopfian group, i.e., every surjective group homomorphism from π 1 (X ) onto itself is an isomorphism.
Given k ∈ N, two crystallographic subgroups of Iso(R k ) are called equivalent if they are conjugated by an affine transformation.The set Crys(k) of equivalence classes of crystallographic subgroups of Iso(R k ) is a finite set thanks to Bieberbach's third Theorem (see Theorem 7.1 in [13]).The following result defines the crystallographic class of X .Proposition 2.7 (Crystallographic class (X )) For i ∈ {1, 2}, let (X , d i , m i ) be an RCD(0, N )-structure on X , and let φ i be a splitting of its lift ( X , di , mi ).Then (φ 1 ) and (φ 2 ) are equivalent as crystallographic subgroups of Iso(R k(X ) ).We denote by (X ) the common equivalence class and call it the crystallographic class of X.
Proof By Bieberbach's second Theorem (see Theorem 4.1 of [13]), two crystallographic subgroups of Iso(R k ) are conjugated by an affine transformation if and only if they are isomorphic (we let k := k(X )).We need to show that (φ 1 ) and (φ 2 ) are isomorphic.Observe that, for i ∈ {1, 2}, we have the following exact sequence of groups: where ι is just the inclusion, (φ i ) is a crystallographic subgroup of Iso(R k ), and K (φ i ) is finite.By Remark 2.5 of [40], K (φ i ) = ι(K (φ i )) is uniquely characterized as the maximal finite normal subgroup of π 1 (X ).In particular, we necessarily have Given k ∈ N, and a crystallographic subgroup of Iso(R k ), the quotient space R k / has the structure of a compact flat orbifold of dimension k, whose orbifold metric d satisfies: where x, y ∈ R k , [x] and [y] are their equivalence classes in R k / , and the infimum is taken over all x ∈ [x] and y ∈ [y].Moreover, equivalent crystallographic groups give rise to orbifolds that are affinely equivalent.
Conversely, given a compact flat orbifold (X , d) of dimension k, the orbifold fundamental group π orb 1 (X ) acts by isometries on the orbifold universal cover, which is R k (the action being discrete and cocompact).Hence, one can associate a crystallographic group to (X , d).Finally, two affinely equivalent flat orbifolds of dimension k have isomorphic orbifold fundamental groups.Hence, by Bieberbach's second Theorem (see Theorem 4.1 in [13]), they give rise to equivalent crystallographic groups (see the introduction of Section 2.1 in [8] for more details and some references).
Therefore, there is a one-to-one correspondence between equivalence classes of crystallographic subgroups of Iso(R k ) and affine equivalence classes of compact flat orbifolds of dimension k.This leads us to the definition of the Albanese class of X .

Definition 2.3 (Albanese class A(X ))
We write A(X ) the set of the affine equivalence classes of compact flat orbifolds determined by (X ), and call it the Albanese class of X .More explicitly, A(X ) is the set of all flat orbifolds (R k(X ) / , d ), where ∈ (X ), and d is defined in equation (2).

Moduli spaces and their topology
In Sect.2.3.1, we will introduce the moduli space of pointed RCD(0, N )-structures on X .Then, in Sect.2.3.2,we will introduce the moduli space of equivariant pointed RCD(0, N )-structures on X .In particular, (based on the equivariant distance introduced by Fukaya and Yamaguchi in [20]), we will introduce the equivariant pmGH topology.

Moduli space of pointed RCD(0,N)-structures
Throughout the paper, we will use the following definition of pointed metric measure spaces.As for m.m.s., there are two distinct notions of isomorphisms between two pointed metric measure spaces.In this paper, we decided to use the following definition (which emphasizes the whole space's metric structure, not only the metric structure of the measure's support).Definition 2.5 Two p.m.m.s.(X 1 , d 1 , m 1 , * 1 ) and (X 2 , d 2 , m 2 , * 2 ) are isomorphic when there is an isomorphism of metric measure spaces φ : Thanks to Theorem 2.7 of [1], and Remark 3.29 of [23], we have the following result.

Theorem 2.3
The Gromov-Hausdorff-Prokhorov distance d GHP (see Section 2.3 of [1]) is a complete separable metric on the set X p of isomorphism classes of pointed metric measure spaces that are locally compact and geodesic.Moreover, d GHP metrizes the pointed measured Gromov-Hausdorff topology (introduced in Definition 27.30 of [39]).
As we will see in Remark 2.2 below, it is possible to realize the pmGH convergence using maps with small distortion.
be a map between metric spaces, we denote: where the supremum is taken over all couples x, y ∈ X .

Remark 2.2 Assume that {X
is a familly of locally compact geodesic p.m.m.s.such that X n → X ∞ in the pmGH topology.Then, Theorem 2.3 implies that there exists a sequence { f n , g n , n } where f n : (X n , * n ) → (X ∞ , * ∞ ) and g n : (X ∞ , * ∞ ) → (X n , * n ) are pointed Borel maps, n → 0, and the following properties are satisfied: Such a sequence is said to realize the convergence of {X n } to X ∞ in the pmGH topology.
We conclude this section by introducing the moduli space of pointed RCD(0, N )structures on X .

Notation 2.3
We introduce the following spaces: (i) RCD p (0, N ) is the set of isomorphism classes of pointed RCD(0, N ) spaces with full support, endowed with the pmGH-topology (seen as a subspace of X p ), (ii) R p 0,N (X ) is the set of all pointed RCD(0, N )-structures on X , (iii) M p 0,N (X ) is the quotient of R p 0,N (X ) by isomorphisms, endowed with the pmGHtopology (seen as a subspace of RCD p (0, N )).
We call M p 0,N (X ) the moduli space of pointed RCD(0, N )-structures on X .123

Moduli space of equivariant pointed RCD(0,N)-structures
First of all, we introduce equivariant pointed RCD(0, N )-structures on X .Here, in comparison with the definition of equivariant metric given by Fukaya and Yamaguchi in [20], both the topological space and the group action are fixed.Definition 2.6 A pointed RCD(0, N )-structure ( X , d, m, * ) on X is called equivariant if π 1 (X ) acts by isomorphisms on ( X , d, m).
We now introduce the space (and moduli space) of equivariant pointed RCD(0, N )structures on X .Notation 2. 4 We introduce the following spaces: (i) R p,eq 0,N ( X ) the set of equivariant pointed RCD(0, N )-structures on X , (ii) M p,eq 0,N ( X ) the quotient space of R p,eq 0,N ( X ) by equivariant ismormophisms.We call M p,eq 0,N ( X ) the moduli space of equivariant pointed RCD(0, N )-structures on X .
To define a topological structure on M p,eq 0,N ( X ) , we start by introducing the equivariant pointed distance on R p,eq 0,N ( X ).Definition 2.8 Let > 0, and, for i ∈ {1, 2}, let Xi = ( X , di , mi , * i ) be an equivariant pointed RCD(0, N )-structure on X .An equivariant pointed -isometry between X1 and X2 is a triple ( f , g, φ) where f : X → X and g : X → X are Borel maps and φ is an isomorphism of π 1 (X ) such that: We define D eq p ( X1 , X2 ) the equivariant pointed distance between X1 and X2 as the minimum between 1/24 and the infimum of all > 0 such that there exists an equivariant pointed -isometry between X1 and X2 .
The following result shows that we can endow M p,eq 0,N ( X ) with a metrizable topology.Proposition 2.8 D eq p induces a metrizable uniform structure on M p,eq 0,N ( X ).Proof See Appendix.
From now on, we endow M p,eq 0,N ( X ) with the topology induced by D eq p , which we call the equivariant pmGH-topology.

Maps between moduli spaces
In Sect.2.4.1, we are going to introduce the lift and push-forward maps.As we will explain at the end of that section, a consequence of Theorem A is that these maps are homeomorphisms and respectively inverse to each other (see Corollary A).Then, in Sect.2.4.2, we will introduce the Albanese map and the soul map associated to X .

Lift and push-forward maps
Thanks to Corollary 2.1, we can define the lift of a pointed RCD(0, N )-structure.Definition 2.9 Let (X , d, m, x) be a pointed RCD(0, N )-structure on X and let x ∈ p −1 (x).We define p * x (X , d, m, x) := ( X , d, m, x), where ( X , d, m) is the lift of (X , d, m).

Remark 2.3 For
). Thanks to Remark 2.3, we can define the lift map associated to X .Definition 2.10 (Lift map) The lift map associated to X is the unique map p * : Thanks to Proposition 2.3, we can define the push-forward of an equivariant pointed RCD(0, N )-structure.Definition 2.11 Let ( X , d, m, x) be an equivariant pointed RCD(0, N )-structure on X .We define p * ( X , d, m, x) as the unique pointed RCD(0, N )-structure on X such that p : ( X , d, m, x) → p * ( X , d, m, x) is a pointed local isomorphism.

Remark 2.4 For
Thanks to Remark 2.4, we can define the push-forward map associated to X .Definition 2.12 (Push-forward map) The push-forward map associated to X is the unique map Thanks to Remark 2.1, we have the following proposition.Proposition 2.9 The lift map p * : M p 0,N (X ) → M p,eq 0,N ( X ) and the push-forward map p * : M p,eq 0,N ( X ) → M p 0,N (X ) are respectively inverse to each other.Observe that Corollary A immediately follows from Proposition 2.9 and Theorem A (which we will prove in Sect.3.1).

Albanese and soul maps
First of all, we introduce the moduli space of flat metrics on the Albanese class A(X ) (introduced in Definition 2.3).This moduli space will act as the codomain of the Albanese map.

Definition 2.13 (M flat (A(X )))
The moduli space of flat metrics on A(X ) is the quotient of A(X ) by isometry equivalence, endowed with the Gromov-Hausdorff distance d GH (see Definition 7.3.10 in [11]).
The following remark will be helpful in the proof of Theorem C. It is also interesting on its own as it gives a more explicit way to see the moduli space M flat (A(X )).
Remark 2.5 Given any element ∈ (X ), the moduli space of flat metrics on A(X ) is isometric to the moduli space of flat metrics on the compact orbifold R k / (endowed with the Gromov-Hausdorff distance), which we denote M flat (R k / ) (see Section 4.2 in [8] for more details on M flat (R k / )).
Thanks to Lemma 2.1, we can define the Albanese and soul maps.Definition 2.14 (Albanese and soul maps) Given an RCD(0, N )-structure (X , d, m) on X , and given a splitting φ of ( X , d, m) with soul (X , d, m), we define: and: The map A : M 0,N (X ) → M flat (A(X )) is called the Albanese map associated to X , and the map is called the soul map associated to X .
We end this section with the following surjectivity result.

Proposition 2.10 The Albanese map associated to X is surjective from
Proof First of all, let (X , d 0 , m 0 ) be a reference RCD(0, N )-structure on X , and let φ 0 be a splitting of its lift ( X, d0 , m0 ) with soul (X 0 , d 0 , m 0 ).Now, let ∈ (X ) and let us show that there is some Since (φ 0 ) ∈ (X ), there is α ∈ Aff(R k ) such that = α (φ 0 )α −1 .Now, let ψ := (id X 0 , α) • φ 0 , and consider the metric measure structure ( d, m) defined as the pull back by ψ of (d 0 × d eucli , m 0 ⊗ L k ).Note that ψ is a homeomorphism, and ( X , d, m) is an RCD(0, N )-structure on X .Now, we are going to show that ( X , d, m) is the lift of some (X , d, m).Thanks to Remark 2.1, it is equivalent to show that π In conclusion, there is an RCD(0, N )-structure (X , d, m) ∈ R 0,N (X ) whose lift is ( X , d, m).By construction, ψ is a splitting of ( X , d, m) with soul (X 0 , d 0 , m 0 ).Moreover, we have seen above that, for every η ∈ π 1 (X ), we have

Proof of Theorem A
First of all, let us introduce the systole associated to an RCD(0, N )-structure on X .Finding a uniform lower bound on the systoles associated to a sequence will be the key to prove Theorem A. Definition 3.1 (Systole of an RCD(0, N )-structure) The systole associated to an RCD(0, N )-structure (X , d, m) on X is the quantity sys(X , d) := inf{ d(η • x, x)}, where the infimum is taken over all point x ∈ X and η ∈ π 1 (X )\{id}.Whenever π 1 (X ) is trivial, we define sys(X , d) := ∞.
The following proposition relates the systole of an RCD(0, N )-structure (X , d, m) on X and the quantity δ(X , d) introduced in Theorem 2.1.(X , d, m) be an RCD(0, N )-structure on X .Then, sys(X , d) = 2δ(X , d), where δ(X , d) is defined in Theorem 2.1.
The next result shows that we can find a positive uniform lower bound on the systoles associated to a converging sequence of RCD(0, N )-structures on X .
Proof First of all, observe that by Theorem 2.1, δ(X , d n ) > 0 for every n ∈ N. In particular, it is sufficient to prove that there exists a constant δ > 0 such that δ(X , d n ) ≥ δ whenever n is large enough.
In particular, we necessarily have ; hence, by the classification Theorem (see Theorem 2, Chapter 2, Section 5 in [34]), (X δ 1 d n , X , p δ 1 d n ) is equivalent to ( X , X , p).In particular, every ball of radius δ 1 in (X , d n ) is evenly covered by p; thus, δ(X , d n ) ≥ δ 1 , which concludes the proof.
The following proposition is a converse to Proposition 3.2; it will be essential to prove the converse implication of Theorem A. Proposition 3.3 Assume that {( X , dn , mn , * n )} converges to ( X , d∞ , m∞ , * ∞ ) in the equivariant pmGH-topology, where, for every n ∈ N ∪ {∞}, ( X , dn , mn , * n ) is an equivariant pointed RCD(0, N )-structure on X .Then 0 < inf n∈N {δ(X , d n )}, where (X , d n , m n ) is the push-forward of ( X , dn , mn ).
Proof We fix a sequence {( fn , gn , φ n , n )} realizing the equivariant pointed convergence.Looking for a contradiction, assume that inf n∈N {sys(X , d n )} = 0. Without loss of generality, we can assume (passing to a subsequence if necessary) that there exist sequences { xn } in X and {γ n } in π 1 (X )\{id} such that dn (γ n xn , xn ) → 0. However, when n is large enough so that dn , which is the contradiction we were looking for.Hence 0 < inf n∈N {sys(X , d n )}; therefore, thanks to Proposition 3.1, we have 0 < inf n∈N {δ(X , d n )}.
We can now prove Theorem A.

Proof of Theorem A, direct implication
Let us prove that { Xn } converges in the equivariant pmGH-topology to X∞ .
Part I: Construction of the realizing sequence { fn , gn , ψ n , n } First of all, we fix a sequence { f n , g n , n } realizing the convergence of {X n } to X ∞ in the pmGH-topology.Then, we define δ := inf n∈N∪{∞} {δ(X , d n )}, which satisfies δ > 0 thanks to Proposition 3.2.By Proposition 3.1, we have μ 0 := inf n∈N∪{∞} , d n )} = 2δ > 0. We define α := δ/2, and we assume that n is large enough so that: . Now, thanks to Theorem 16 of [31] (and the construction in its proof), there exists a triple ( fn , gn , ψ n ) such that: • for every x ∈ X , we have dn ( gn • fn ( x), x) ≤ n and d∞ ( fn • gn ( x), x) ≤ n , • for every x ∈ X and η ∈ π 1 (X ), we have fn Moreover, using inequality 3, Theorem 16 of [31] assures that, for every x, ỹ ∈ X , we have: and: We fix C > 0 such that C + 3/α ≤ C 2 , and we define n := C n .When n is large enough so that n ≤ n , we have: ( fn , gn , ψ n , n ) satisfies point (i) to (iv) of Definition 2.8 w.r.t.Xn and X∞ .(4) Let us prove that, when n is large enough, fn and gn are Borel maps.Let x ∈ X , and let r < δ/3.Thanks to Proposition 2.2 and property 4, we easily get: when n is large enough so that δ/3 + 4 n < δ/2 < −1 n , and where x := p( x).However, f n is a Borel map, and p is continuous; therefore f −1 n ( B∞ ( x, r )) is a Borel subset of X .We have shown that when n is large enough, the pre-image by fn of balls of radius r < δ/3 are Borel subsets of X .Therefore, for n large enough, fn is a Borel map, and the same is true for gn with the same procedure.

Part II: Measured convergence
Our goal here is to prove that (making n larger if necessary but keeping n → 0), we have: max{d This implied by the fact that, {d {R} P ( fn * mn , m∞ )} and {d {R} P ( gn * , mn )} converge to 0 as n goes to infinity, for every R > 0.
First of all, observe that lim n→∞ d {R} P ( fn * mn , m∞ ) = 0 for every R > 0 if and only if { fn * mn } converges to m∞ in the weak- * topology.Then, note that the space M loc ( X , d∞ ) of Radon measures on ( X , d∞ ) endowed with the weak- * topology is metrizable (see Theorem A2.6.III in [17]).Hence, it is sufficient to show that any subsequence of { fn * mn } admits a subsequence converging to m∞ .Without loss of generality (reindexing the sequence if necessary), let us just show that { fn * mn } admits a subsequence converging to m∞ .
First, let us show that { fn * mn } is precompact, which is implied by the uniform boundedness of for every R > 0 (see Theorem A2.6.IV and Theorem A2.4.I in [17]).We define Observe that r 0 is positive thanks to Proposition 3.2, and M is finite since {m n (X )} converges to m ∞ (X ), which is finite.Thanks to point (v) of Proposition 2.2, we have mn (B dn (r 0 )) = m n (B d n (r 0 )) ≤ M, for every n ∈ N.Then, thanks to property 4, , for every R > 0, and n sufficiently large.Now, consider the following two cases: when n is sufficiently large, • if R > r 0 /2, thanks to Bishop-Gromov inequality for RCD(0, N ) spaces (see Theorem 6.2 in [6]), we get fn * mn (B d∞ (R)) ≤ mn (B dn (2R)) ≤ M(2R/r 0 ) N , when n is sufficiently large.
In particular, for every R > 0, the sequence { fn * mn (B d∞ (R))} is uniformly bounded; hence { fn * mn } is precompact.Now, passing to a subsequence if necessary, we can assume that { fn * mn } is converging to some m ∈ M loc ( X , d∞ ).Let us show that m = m∞ .Note that it is sufficient to prove that, for every x ∈ X and 0 < r < r 0 , we have m(B d∞ ( x, r )) = m∞ (B d∞ ( x, r )); since small balls generate the Borel σ -algebra of X .
First, observe that, for every n ∈ N, we have mn In addition, m is positive since {m n (B d n (r 0 ))} is a sequence of positive numbers converging to m ∞ (B d ∞ (r 0 )), which is positive.Therefore, { Xn } is a sequence of pointed RCD(0, N ) spaces with measures uniformly bounded from below; hence (thanks to 7.2 in [23]), any limit point in the pmGH-topology is a full support RCD(0, N space.However, the sequence converges in the pmGH-topology to ( X , d∞ , m, * ∞ ).Thus, ( X , d∞ , m, * ∞ ) is a full support RCD(0, N ) space.In particular, thanks to Theorem 30.11 in [39], we have m(∂ B d∞ ( x, R)) = 0 for every R > 0 and x ∈ X .Hence, thanks to Proposition A2.6.II in [17], for every R > 0 and x ∈ X we have: Now, let x ∈ X and 0 < r < r 0 , and let us show that we have m(B d∞ ( x, r )) = m∞ (B d∞ ( x, r )).First, when n is large enough so that r ≤ −1 n , we can use property 4 to get: In particular, defining A := m(B d∞ ( x, r )) and using equation 5, we have: Moreover, when n is large enough, we have r + 2 n < r 0 < δ/2; hence, point (v) of Proposition 2.2 implies: where x := p( x).Now, observe that when n is large enough so that r + 4 n ≤ −1 n , we can use property 4 to get: In particular, for every η > 0, we have: However, since { f n * m n } converges to m ∞ , and since X ∞ is a full support RCD(0, N ) space, we can apply Theorem 30.11 in [39] and Proposition A2.3.II in [17] to get: Hence, for every η > 0, we have: and, letting η go to 0, have m(B d∞ ( x, r )) = m ∞ (B d ∞ (x, )) = m∞ (B d∞ ( x, r )) (using r < r 0 < δ/2 for the last equality).Therefore { fn * mn } converges to m∞ .

Proof of Theorem A, converse implication
Let { fn , gn , φ n , n } be a sequence realizing the convergence of { Xn } to X∞ in the equivariant pmGH-topology.Thanks to the equivariant requirement, there exists pointed Borel maps Let us fix x ∈ X and x ∈ p −1 (x).Observe that d n (g n ( f n (x)), x) = inf{ dn ( ỹ, x)}, where the infimum is taken over all ỹ ∈ X such that p( ỹ) = g n ( f n (x)).However, we have p( gn ( fn ( x))) = g n ( f n (x)).Therefore, we have d n (g n ( f n (x)), x) ≤ dn ( gn ( fn ( x)), x) ≤ n .The same argument shows that d ∞ ( f n (g n (x)), x) ≤ n .
Let y i ∈ X (i ∈ {1, 2}) and let ỹi such that p( ỹi ) = y i and d∞ ( ỹ1 and observe that, since p( gn ( ỹi )) = g n (y i ), we have: Then, let Let us show that and observe that thanks to inequality 6, we have { Xn } to X∞ ) in the pmGH-topology (resp. in the equivariant pmGH-topology), such that p • fn = f n • p and p • gn = g n • p. Finally, for every n ∈ N, we define: , and l S n := p X n • l n (•, 0).The main difficulty of the argument will be to prove that k n and l n almost split.More precisely, we will show that k n (k S n , k R n ) and l n (l S n , l R n ) (where we will give a precise meaning to ).Then, we will deduce property 10 and property 11 from that.
First of all, we prove that {Diam(X n , d n )} is bounded.
Proof Looking for a contradiction, let us suppose that lim sup n→∞ Diam(X n , d n ) = ∞.Passing to a subsequence if necessary, we can assume that Diam(X n , d n ) > 2 n+1 , for every n ∈ N. Hence, there are sequences {x n } and {z n } such that, for every n ∈ N, we have x n , z n ∈ X n , and For every n ∈ N, let γ n : [−2 n , 2 n ] → X n be a minimizing geodesic parametrized by arc length from x n to z n , and let us denote γn := (γ n , 0).Thanks to Proposition 2.4, there exists η Then, let us define βn := η γn , and denote β n := p X n ( βn ), and in the pmGH-topology.Therefore, thanks to Arzelà-Ascoli Theorem (see Proposition 27.20 in [39]), we can assume (passing to a subsequence if necessary) that {k n • βn } converges locally uniformly to an isometric embedding β : R → X ∞ × R k .However, (X ∞ , d ∞ ) is compact; thus, applying Lemma 1 of [32], there exists a, b ∈ R k and y ∞ ∈ X ∞ such that, for every t ∈ R, β(t) = (y ∞ , at + b), and a = 1.Now, we define y n := β n (0) and, for u ∈ R k , n (u) := (y n , u) ∈ X n × R k .Observe that { n } is a sequence of isometric embeddings such that, for every n ∈ N, we have n (0) ∈ B X n ×R k (( * n , 0), D).Therefore, thanks to Arzelà-Ascoli Theorem (see Proposition 27.20 in [39]), we can assume (passing to a subsequence if necessary) that {k n • n } converges locally uniformly to an isometric embedding : R k → X ∞ ×R k .Moreover, since X ∞ is compact, we can easily deduce from Lemma 1 of [32] that there ). Therefore: Therefore, we have 0 = (1+ c 2 ) 1/2 0, which is the contradiction we were looking Thanks to Proposition 3.4, we can introduce the following notations.
Notation 3.1 We denote D := sup n∈N∪∞ {Diam(X , d n )} < ∞ (finiteness being granted by the convergence of {X , d n } to (X , d ∞ ) in the GH-topology).We also denote Our first goal will be to obtain a convergence result on the following "splitting quantities".Notation 3.2 (Splitting quantities) Given n ∈ N and R > 0, we define: The following next two technical lemmas will be our main ingredients in the proof of the convergence result of the splitting quantities.Lemma 3.1 Let {y n } be a sequence such that, for every n ∈ N, y n ∈ X n .For every n ∈ N, and t ∈ R k , we define n : t ∈ R k → (y n , t) ∈ X n ×R k .Then, the sequence of maps {k n • n : R k → X ∞ × R k } admits a subsequence converging locally uniformly to a map : R k → X ∞ × R k .Moreover, for any such limit , there exists y ∞ , and Proof Observe that, for every n ∈ N, n is an isometric embedding that satisfies Therefore, applying Arzelà-Ascoli Theorem (see Proposition 27.20 in [39]) as in the proof of Proposition 3.4, we can assume without loss of generality that {k n • n } converges locally uniformly to an isometric embedding : R k → X ∞ × R k .Moreover, using Lemma 1 of [32], there exist φ ∈ Iso(R k ) and y ∞ ∈ X ∞ such that (t) = (y ∞ , φ(t)), for every t ∈ R k .To conclude, we need to show that φ(0) = 0. First, observe that: whenever n is large enough (so that D ≤ −1 n ), and where and observe that: when n is large enough (so that (D 2 + |t| 2 ) ≤ −1 n ), and where n : Hence, combining the two inequalities above, we obtain: In conclusion, φ(0) = 0. Lemma 3.2 Let {y n } and {z n } be sequences such that, for every n ∈ N, y n , z n ∈ X n .For every n ∈ N and t ∈ R k , we define n (t) := (y n , t) and n (t) := (z n , t).Assume that (passing to a subsequence if necessary), the sequences of maps {k n • n } and Proof Looking for a contradiction, let us suppose that φ = ψ.In that case, there exists where , and when n is large enough (so that D ≤ −1 n ).Now, observe that lim n→∞ u n = 0; therefore, passing to the limit in inequality 12, we have d We can now state the convergence result on the splitting quantities.
Looking for a contradiction, we assume that lim n→∞ α(n, R) = 0. Passing to a subsequence if necessary, there exist > 0, and sequences {y n } and {t n } such that: In particular, we have: Now, applying Lemma 3.1 and Lemma 3.2, we can also assume that {k n • n } converges locally uniformly to = (z ∞ , φ), where n (s) := ( * n , s), and z ∞ ∈ X ∞ .Thus, we have: Hence, using equations 14 and 15, we have lim Part II: lim n→∞ β(n, R) = 0 Looking for a contradiction, we assume that lim n→∞ β(n, R) = 0. Passing to a subsequence if necessary, there exist > 0, and sequences {y Observe that {t n } is bounded and X ∞ is compact; therefore, we can assume that t n → t and y Then, applying Lemma 3.1 and Lemma 3.2, we can also assume that {k n • n } converges locally uniformly to = (z ∞ , φ), where n (u) := ( * n , u), and z ∞ ∈ X ∞ .Moreover, (0 Finally, observe that equations 17 and 18 contradict inequality 16, which concludes the proof.
The continuity of the soul map is a consequence of the following proposition, which gives us property 10 as a corollary.

Proposition 3.5 The sequence {k
We are going to show that there exists a map R : N × R ≥0 → R ≥0 such that for every R > 0: Then we will prove that {k R n * L k } converges to L k for the weak- * topology.Let R > 0, and let Hence, thanks to Lemma 3.3, we get d ), for n large enough.Therefore, we get point (ii) if we set R (n, R) := 2( n + α(n, 2R) + β(n, 2R)).Moreover, thanks to Lemma 3.3, we have lim n→∞ R (n, R) = 0, for every R > 0. Now, given t 1 , t 2 ∈ R k such that |t 1 |≤ R and |t 2 |≤ R, we define: Using k S n ( * n ) = * ∞ , we get: Hence, for n large enough, we have: In particular, this implies Dis(k R n |B R k (0,R) ) ≤ 2α(n, R) + n .Moreover, k n and l n playing symmetric roles, we also have Dis(l R n |B R k (0,R) ) ≤ 2β(n, R) + n .We replace R (n, R) by R (n, R) + 2α(n, R) + 2β(n, R) + n .This concludes the proof of (i), (ii), and (iii) (thanks to Lemma 3.3).
Let us prove that {k R n * L k } converges to L k for the weak- * topology.Here, the strategy will be the same as in the proof of A. More precisely, the weak- * topology on M (R k ) is metrizable; therefore, it is equivalent to prove that every subsequence of {k R n * L k } admits a subsequence converging to L k in the weak- * topology.Let us just prove that {k R n * L k } admits a subsequence converging to L k in the weak- * topology (the proof for a subsequence being exactly the same).First of all, observe that thanks to Lemma 3.1, we can assume (passing to a subsequence if necessary) that {t ∈ In particular, we have: where Rn : then n is sufficiently large).Hence, whenever n is large enough, we have: where n converges locally uniformly to φ; thus ν n → 0. In particular, since Let y n ∈ X n , and observe that d In conclusion, we have: Since k n and l n play symmetric roles, we also have: for y ∞ ∈ X ∞ .Observe that, thanks to Lemma 3.3, we have n→∞ S n = 0. Now, let y 1 , y 2 ∈ X n , and define: Then, we have: Hence, we finally get (for n large enough): In particular, we have Dis(k S n ) ≤ 2α(n, 0) + n .Then, k n and l n playing symmetric roles, we also have Dis(l S n ) ≤ 2β(n, 0) + n .Finally, replacing S n by S n + n + 2 max{α(n, 0), β(n, 0)}, and applying Lemma 3.3, we have S n → 0; therefore, using inequalities 19 and 20, we can conclude that {X n , d n } converges to (X ∞ , d ∞ ) in the GH-topology.Now let us prove that {k S n * m n } converges to m ∞ in the weak- * topology.As we've seen in Part I, it is sufficient to show that {k S n * m n } admits a subsequence converging to m ∞ in the weak- * topology (the weak- * topology being metrizable).
First, let us show that {k S n * m n } is precompact in the space M(X ∞ ) of Radon measure on X ∞ , which is implied by the uniform boundedness of the sequence {m n (X n )}.Let us fix r 0 ∈ (0, δ/2), where δ := inf n∈N∪{∞} {δ(X , d n )} (δ being positive thanks to Proposition 3.2).Then, using point (v) of Proposition 2.2 and Theorem 2.1, observe that In particular, for every n ∈ N, we can apply Bishop-Gromov inequality (see Theorem 6.2 in [6]), and get converges to m ⊗ L k in the weak- * topology.In addition, thanks to Theorem A, Then, proceeding as in Part I of the proof, we obtain: when n is sufficiently large and where R := (D 2 + R 2 ) 1/2 .In particular, we have: where ω φ is the modulus of uniform continuity associated to φ.Then, thanks to Lemma 3.3, we have lim n→∞ ω φ (α(n, 2 R)) = 0. Thus, for every φ ∈ C c (X ∞ ×R k ), we have: Inspired by the proof of Theorem 5.4 in [37], we introduce the following "shrunk" metrics.
The following lemma shows that the "shrunk" metrics associated to the sequence {X , d n , m n } are close to the corresponding Albanese varieties.Proof First of all, observe that there exists a continuous map a :
We conclude this section with the following proposition, which states the continuity of the Albanese map by proving property 11.In conclusion, we have: However, thanks to Lemma 3.5, we have lim n,m→∞ 2 n + (n, m, D)+ (n, m, D) = 0, which concludes the proof.

Proof of Theorem C
The proof of Theorem C is inspired by the proof of Theorem 1.1 in [37] and uses some of the computations realized in [29].First of all, using Theorem B, we are going to prove the following result.
To conclude the proof, we apply the same idea, using the fact 3 (T 4 ) ⊗ Q Q, and π 5 (T 5 ) ⊗ Q Q (see Proposition 5.5 of [37]).
Using the same argument for g, we can conclude that point (iii) of Definition 2. 8   p,eq 0,N ( X ) .Thanks to the fact that D eq p is well defined on M p,eq 0,N ( X ), symmetric, nonnegative, and satisfies the modified triangle inequality 28, we can easily check the axioms introduced p.141 of [9].Therefore, B is a fundamental system of neighborhood of a uniform structure on M p,eq 0,N ( X ).
Part III: Metrizable uniform structure We have seen that D eq p induces a Hausdorff uniform structure on M p,eq 0,N ( X ).Moreover, B is a countable system of fundamental neighborhoods for this uniform structure.Therefore, thanks to Proposition 2 p.126 of [10], there exists a distance d : M p,eq 0,N ( X ) × M p,eq 0,N ( X ) → [0, +∞] such that d induces the same uniform structure as D eq p .Observe that we can assume, without loss of generality, that d is finite (replacing d by min{1, d} if necessary), which concludes the proof.

Definition 2 . 1
Given δ > 0 and (X , d, m) an RCD(0, N )-structure on X , the δ-cover associated to (X , d, m) is the regular covering p δ d : X δ d → X associated to the open cover U(δ, d) consisting of balls of radius δ for the distance d (see Proposition 2.1).
(i) Noetherian groups (every subgroup is finitely generated) are Hopfian groups.(ii) If H is a normal subgroup of G such that both H and G/H are Noetherian, then G is Noetherian.(iii) Finite groups are Noetherian.(iv) Finitely generated abelian groups are Noetherian.
for every m ∈ N, and n ∈ N ∪ {∞} (where D is defined in Notation 3.1).

1 P
Applying the same argument, we also get d[4( 12 + 23 )] −(g * m3 , m1 ) ≤ 4( 12 + 23 ).Therefore, point (v) of Definition 2.8 is satisfied.This concludes the proof of the modified triangle inequality 28.Part II: Hausdorff uniform structure Let B := {{D eq p ≤ 2 −n }, n ∈ N} ⊂ P M p,eq 0,N ( X ) × M Then {X n } n∈N converges to X ∞ in the pmGH topology if and only if { Xn } n∈N converges to X∞ in the equivariant pmGH topology (introduced in Sect.2.3.2).Note that, since X is compact, it is also possible to formulate Theorem A as follows (forgetting about the reference points in the base space):Assume that for every n dn , mn , * n ) is the associated pointed lift, where * n is any point in p −1 ( * n ).
and |t n | ≤ R.Moreover, since {t n } is bounded, we can assume that t n → t.Now, applying Lemma 3.1, and passing to a subsequence if necessary, we can assume that {k n • n } converges locally uniformly to = (y ∞ , φ), where n (s) : and n (u) := (y n , u), u ∈ R k .Applying Lemma 3.1, we can assume (passing to a subsequence if necessary) that {k n • n } converges locally uniformly to (z ∞ , φ), where z ∞ ∈ X ∞ and φ ∈ O k (R).Observe that {s n } is bounded since {(y ∞ , t n )} converges.Therefore, we can assume that s n → s.However, we have lim n→∞ k n where D is defined in Notation 3.1.In conclusion, {m n (X n )} is uniformly bounded; thus {k S n * m n } is precompact in the weak- * topology.Now, passing to a subsequence if necessary, we can assume that {k S n * m n } converges to some Radon measure m on X ∞ .We need to prove that m Now, note that, by definition of d (φ n ) , we have d (φ n ) (a(y), a(z)) ≤ d eucli (t y , t z ); in particular:0 ≤ d n,m (y, z) − d (φ n ) (a(y), a(z)).(21)Noticethat, by definition of d (φ n ) , there exists η ∈ π 1 (X ) that d (φ n ) (a(y), = d eucli (t y , t 2 ), where t 2 2 n,m (y, z) = 2 −2m d 2 n (y, z) + d 2 eucli (t y , t z ).