An upper bound on the revised first Betti number and a torus stability result for RCD spaces

We prove an upper bound on the rank of the abelianised revised fundamental group (called"revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating that the space is $\lfloor N \rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $\lfloor N \rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-H\"older homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-\delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).

Let us start by recalling that an RCD * (K, N) space is a (possibly non-smooth) metric measure space (X, d, m) with dimension bounded above by N ∈ [1, ∞) and Ricci curvature bounded below by K ∈ R, in a synthetic sense (see Section 2.3 for the precise notions and the corresponding bibliography). The class of RCD * (K, N) spaces is a natural non-smooth extension of the class of smooth Riemannian manifolds of dimension ≤ N and Ricci curvature bounded below by K ∈ R, indeed: • It contains the class of smooth Riemannian manifolds of dimension ≤ N and Ricci curvature bounded below by K ∈ R; • It is closed under pointed measured Gromov-Hausdorff convergence, so Ricci limit spaces are examples of RCD * (K, N) spaces; • It includes the class of ⌊N⌋-dimensional Alexandrov spaces with curvature bounded below by K/(⌊N⌋− 1), the latter being the synthetic extension of the class of smooth ⌊N⌋-dimensional Riemannian manifolds with sectional curvature bounded below by K/(⌊N⌋ − 1); • In contrast to the class of smooth Riemannian manifolds, it is closed under natural geometric operations such as quotients, foliations, conical and warped product constructions (provided natural assumptions are met); • Several fundamental comparison and structural results known for smooth Riemannian manifolds with Ricci curvature bounded below and for Ricci limits have been extended to RCD * (K, N) spaces.
It was proved by Wei and the second named author [MW19] (after Sormani-Wei [SW01,SW04b]) that an RCD * (K, N) space (X, d, m) admits a universal cover ( X, d X , m X ), which is an RCD * (K, N) space as well.
The group of deck transformations on the universal cover is called revised fundamental group of X and denoted byπ 1 (X) (see Section 2.6.1 for the precise definitions and basic properties).
We next discuss the main results of the present paper. Let (X, d, m) be a compact RCD * (K, N) space and letπ 1 (X) be its revised fundamental group. Set H := [π 1 (X),π 1 (X)] and Γ :=π 1 (X)/H respectively the commutator and the abelianised revised fundamental group. As a consequence of Bishop-Gromov volume comparison, Γ is finitely generated (see Proposition 2.25, after Sormani-Wei [SW04a]) and thus it can be written as Γ = Z s × Z s 1 p 1 × · · · × Z s l p l . We define the revised first Betti number of (X, d, m) as b 1 (X) := rank(Γ) = s.
The goal of the paper is two-fold: • First, we prove an upper bound for the revised first Betti number of a compact RCD * (K, N) space, generalising to the non-smooth metric measure setting a classical result of M. Gromov [Gro81] and S. Gallot [Gal83] originally proved for smooth Riemannian manifolds with Ricci curvature bounded below. • Second, we prove a torus stability/almost rigidity result, roughly stating that if (X, d, m) is a compact RCD * (−ε, N) space with b 1 (X) = ⌊N⌋, then a finite cover must be measured Gromov The upper bound of Theorem 1.1 is sharp, as a flat ⌊N⌋-dimensional torus T ⌊N⌋ , is an example of an RCD * (0, ⌊N⌋) space (thus of an RCD * (−ε, N) space for any ε > 0) saturating the upper bound b 1 (T ⌊N⌋ ) = ⌊N⌋.
In order to state the second main result, let us adopt the standard notation ε(δ|N) to denote a real valued function of δ and N satisfying that lim δ→0 ε(δ|N) = 0, for every fixed N. Let us also recall that (see Section 2.5 for more details and for the relevant bibliography): • We say that (X, d, m) has essential dimension equal to N ∈ N if m-a.e. x has a unique tangent space isometric to the N-dimensional Euclidean space R N ; • We say that (X, d, m) is N-rectifiable as a metric measure space for some N ∈ N if there exists a family of Borel subsets U α ⊂ X and charts ϕ α : U α → R N which are bi-Lipschitz on their image such that m(X \ α U α ) = 0 and m U α ≪ H N U α , where H N denotes the N-dimensional Hausdorff measure.
(1) Then (X, d, m) has essential dimension equal to ⌊N⌋ and it is ⌊N⌋-rectifiable as a metric measure space.
(2) There exists a finite cover (X ′ , d X ′ , m X ′ ) of (X, d, m) which is ε(δ|N)-mGH close to a flat torus of dimension ⌊N⌋. The torus stability above should be compared with the torus rigidity below, proved by Wei and the second named author [MW19], extending to the non-smooth RCD * (0, N) setting a classical result of Cheeger-Gromoll [CG72]. See also Gigli-Rigoni [GR18] for a related torus rigidity result, where the maximality assumption on the rank of the revised fundamental group is replaced by the maximality of the rank of harmonic one forms (recall that the rank of the space of harmonic one forms coincides with the first Betti number in the smooth setting).
1.1. Outline of the arguments and organisation of the paper. Our first goal will be to establish the Gromov-Gallot's upper bound on b 1 (X) stated in Theorem 1.1. To that aim: • Let (X, d, m) be a compact RCD * (K, N) space. If N = 1 then all the results hold trivially (see Remark 2.7.1). So we assume that N ∈ (1, ∞); • Let ( X, d X , m X ) be the universal cover of (X, d, m). Recall that ( X, d X , m X ) is an RCD * (K, N) space as well, and the revised fundamental groupπ 1 (X) acts on ( X, d X , m X ) by deck transformations (actuallȳ π 1 (X) can be identified with the group of deck transformations on X); • Let H = [π 1 (X),π 1 (X)] be the commutator ofπ 1 (X) and consider the quotient spaceX = X/H.X inherits a natural quotient metric measure structure from X, denoted by (X, dX, mX), which satisfies the RCD * (K, N) condition as well (see Corollary 2.26). Moreover (X, dX, mX) is a covering space for (X, d, m), with fibres of countable cardinality (corresponding to Γ :=π 1 (X)/H); • We will also consider X ′ := X/Γ ′ , where Γ ′ Z b 1 (X) is a suitable subgroup of Γ. More precisely, fix a pointx ∈X; extending a classical argument of Gromov to the non-smooth RCD setting, one can construct Γ ′ < Γ isomorphic to Z b 1 (X) such that the distance betweenx and any element in Γ ′x is bounded above and below uniformly in terms of diam(X) (see Lemma 3.2 for the precise statement). The quotient space (X ′ , d X ′ , m X ′ ) still satisfies the RCD * (K, N) condition, it is a covering space for (X, d, m), with fibres of finite cardinality (corresponding to the index of Γ ′ in Γ).
After the above constructions, a counting argument combined with Bishop-Gromov's volume comparison Theorem in (X, dX, mX) will give Theorem 1.1 at the end of Section 3.
In order to show Colding's torus stability for RCD * (−δ, N) spaces (i.e. Theorem 1.2), in Section 4 we will construct ε-mGH approximations from large balls inX to balls of the same radius in the Euclidean space R ⌊N⌋ (see Theorem 4.1 for the precise statement). This is achieved by an inductive argument with ⌊N⌋ steps: in each step we obtain that a ball inX is mGH close to a ball in a product R n × Y, where Y is an RCD * (0, N − n) space. In order to prove the inductive step and pass from n to n + 1, we show that for δ > 0 small enough, Y must have large diameter, so that the almost splitting theorem applies to Y. Therefore, we get a mGH approximation from a ball inX into R n+1 × Y ′ . The diameter estimate for Y relies on the volume counting argument described in the previous paragraph and contained in Section 3.
The approach above is inspired by Colding's paper [Col97], however there are some substantial differences: indeed Colding's inductive argument is based on the construction of what are now known as δ-splitting maps, while we only use ε-mGH approximations and the almost splitting theorem; moreover the non-smooth RCD * setting, in contrast to the smooth Riemannian framework, poses some challenges at the level of regularity, of global/local structure, and of topology. Below we briefly sketch the main lines of arguments; the expert will recognise the differences from [Col97].
The existence of ε-mGH approximations into the Euclidean space yields the first claim of Theorem 1.2: for δ > 0 small enough, (X, d, m) has essential dimension equal to ⌊N⌋, it is ⌊N⌋-rectifiable as a metric measure space and moreover, if N is an integer, the measure coincides with the Hausdorff measure H N , up to a positive constant. This will be proved in Theorem 5.1 by combining Theorem 4.1 with an ε-regularity result by Naber and the second named author [MN19], revisited in the light of the constancy of dimension of RCD * (K, N) spaces by Brué-Semola [BS20] and a measure-rigidity result by Honda [Hon20] for noncollapsed RCD * (K, N) spaces.
When {(X i , d i , m i )} i∈N is a sequence of spaces as in the assumptions of Theorem 1.2 with δ i ↓ 0, Theorem 4.1 yields pmGH convergence for (X i , dX i , mX i ) to the Euclidean space of dimension ⌊N⌋. Then by taking the subgroups Z ⌊N⌋ Γ ′ i < Γ i :=π 1 (X i )/H i already considered above (i.e. the ones constructed in Lemma 3.2, with k = 3) and using equivariant Gromov-Hausdorff convergence (introduced by Fukaya [Fuk86] and further developed by Fukaya-Yamaguchi [FY92]), we deduce GH-convergence of (a non re-labeled subsequence of) X ′ i :=X i /Γ ′ i to a flat torus of dimension ⌊N⌋. This will show the second claim of Theorem 1.2 (see Proposition 6.2 for more details).
When N is an integer, the measure of X ′ i coincides with H N (up to a constant), thanks to the aforementioned result by Honda [Hon20]. This fact allows to apply Colding's volume convergence for RCD spaces proved by De Philippis-Gigli [DPG18] and get that the GH-convergence obtained above can be promoted to mGH-convergence of X ′ i to a flat torus. A recent result by Kapovitch and the second named author [KM21] (which builds on top of Cheeger-Colding's metric Reifenberg theorem [CC97]) states that for N ∈ N, if a non-collapsed RCD * (K, N) space is mGH-close enough to a compact smooth N-manifold M, then it is bi-Hölder homeomorphic to M. This implies that for δ > 0 small enough as in Theorem 1.2, X ′ :=X/Γ ′ is bi-Hölder homeomorphic to a flat torus and thusX is locally (on arbitrarily large compact subsets) bi-Hölder homeomorphic to R N . In order to conclude the proof of the third claim of Theorem 1.2, we show that Γ is torsion free, yielding that Γ Z N and thus X =X/Γ is bi-Hölder homeomorphic to a flat torus. This last step uses the classical Smith's theory of groups of transformations with finite period.
The paper is organised as follows. Section 2 is devoted to recall previous results about RCD spaces, covering spaces and pointed Gromov-Hausdorff convergence (measured and equivariant) that are used in the rest of the paper. In particular, we show that a metric measure space (X, d, m) is RCD * (K, N) if and only if any of its regular coverings with countable fibre is an RCD * (K, N) space as well. This is essential since in our proofs we often use properties of RCD * spaces on the coverings X,X and X ′ of X. Section 3 contains the proof of the upper bound for the revised first Betti number and its consequences. In Section 4, we construct by induction ε-mGH approximations between large balls in the coveringX and balls in Euclidean space of dimension b 1 (X) = ⌊N⌋. Section 5 is devoted to proving the ⌊N⌋ rectifiability, i.e. the first claim of Theorem 1.2. In Section 6, we conclude the proof of Theorem 1.2 by first showing that X ′ is GH-close to a flat torus T N and then obtaining that, for integer N, X ′ is bi-Hölder homeomorphic to T N and X = X ′ . In the appendix we construct two explicit mGH-approximations that are used in Section 4.
Acknowledgments. I.M. and R.P. wish to thank the Institut Henri Poincaré for its hospitality in July 2019 where they met to work on this project. A.M. is supported by the European Research Council (ERC), under the European's Union Horizon 2020 research and innovation programme, via the ERC Starting Grant "CURVATURE", grant agreement No. 802689. R.P. wishes to thank the Mexican Math Society and the Kovalevskaya Foundation for the travel support received in November 2018 to visit I.M. the Summer of 2018. She also wants to thank the ANR grant: ANR-17-CE40-0034 "Curvature bounds and spaces of metrics" for the support to travel within Europe to visit I.M. in July 2019. The authors thank Daniele Semola for carefully reading a preliminary version of the manuscript and for his comments.

BACKGROUND
In this section we recall some fundamental notions about convergence of metric measure spaces and about metric measures spaces with a synthetic lower bound on the Ricci curvature which will be used in the paper.
2.1. Metric measure spaces and pointed metric measure spaces. A metric measure space (m.m.s. for short) is a triple (X, d, m) where (X, d) is a complete and separable metric space and m is a locally finite non-negative complete Borel measure on X, with X = supp(m) and m(X) > 0. A pointed metric measure space (p.m.m.s. for short) is a quadruple (X, d, m,x) where (X, d, m) is a m.m.s. andx ∈ X is a given reference point. Two p.m.m.s. (X, d, m,x) and (X ′ , d ′ , m ′ ,x ′ ) are said to be isomorphic if there exists an isometry Recall that (X, d) is said to be • proper if closed bounded sets are compact; • geodesic if for every pair of points x, y ∈ X there exists a length minimising geodesic from x to y; As we will recall later in this section, the synthetic Ricci curvature lower bounds used in the paper (i.e. CD * (K, N) for some K ∈ R, N ∈ [1, ∞)) imply that (X, d) is proper and geodesic (see Remark 2.8.1).
Definition 2.1 (Definition of pmGH convergence via pmGH approximations). Let (X n , d n , m n ,x n ), n ∈ N ∪ {∞}, be a sequence of p.m.m.s. We say that (X n , d n , m n ,x n ) converges to (X ∞ , d ∞ , m ∞ ,x ∞ ) in the pmGH sense if for any ε, R > 0 there exists N(ε, R) ∈ N such that, for each n ≥ N(ε, R), there exists a Borel map f R,ε n : B R (x n ) → X ∞ satisfying: • ( f R,ε n ) ♯ (m n B R (x n )) weakly converges to m ∞ B R (x ∞ ) as n → ∞, for a.e. R > 0. The maps f R,ε n : B R (x n ) → X ∞ are called ε-pmGH approximations. If we do not require the maps f R,ε n to be Borel, nor the last item to hold, we say that the maps f R,ε n are ε-pGH approximations and that the sequence converges in pointed Gromov-Hausdorff (pGH) sense.
We next define equivariant pointed Gromov-Hausdorff (EpGH) convergence. To this aim, given a metric space (X, d), we endow its group of isometries Iso(X) with the compact-open topology. In this case, it is known that the compact-open topology is equivalent to the topology induced by uniform convergence on compact sets (see for example [Mun00, Theorem 46.8]). When X is proper, a sequence ( f n ) n∈N of isometries of X converges to f in the compact-open topology if and only if ( f n ) n∈N converges to f point-wise on X.
Remark 2.1.1. Given any x 0 ∈ X, denote Let M p eq be the set of quadruples (X, d,x, Γ), where (X, d,x) is a proper pointed metric space and Γ ⊂ Iso(X) is a closed subgroup of isometries. Define the set Γ(r) = {γ ∈ Γ | γ(x) ∈ B r (x)}. We are now in position to define equivariant pointed Gromov-Hausdorff convergence for elements of M p eq . Definition 2.2. Let (X n , d n ,x n , Γ n ) ∈ M p eq , n = 1, 2. An ε-equivariant pGH approximation is a triple of functions ( f, φ, ψ), (4) For all γ 1 ∈ Γ 1 (ε −1 ) such that x, γ 1 x ∈ B ε −1 (x 1 ), it holds Note that we do not assume f to be continuous, nor φ and ψ to be homeomorphisms.
Definition 2.3. A sequence {(X n , d n ,x n , Γ n )} n∈N of spaces in M p eq converges in the equivariant pointed Gromov-Hausdorff (EpGH for short) sense to (X ∞ , d ∞ ,x ∞ , Γ ∞ ) ∈ M p eq if there exist ε n -equivariant pGH approximations between (X n , d n ,x n , Γ n ) and (X ∞ , d ∞ ,x ∞ , Γ ∞ ) such that ε n → 0, as n → ∞. such that {(X n , d n ,x n )} n∈N converges in the pointed Gromov-Hausdorff sense to (X ∞ , d ∞ ,x ∞ ). Then there exist Γ ∞ a closed subgroup of isometries of X ∞ and a subsequence, {(X n j , d n j ,x n j , Γ n j )} j ∈ M p eq , that converges in equivariant pointed Gromov-Hausdorff For a closed subgroup Γ in Iso(X) and x ∈ X, let Γx ⊂ X denote the orbit of x under the action of Γ. The space of orbits is denoted by X/Γ. Let (1) It is a standard fact that d X/Γ defines a distance on X/Γ. Indeed, the equivalence between convergence in compact-open topology and point-wise convergence in X implies that the orbits of Γ are closed in x. Then consider Γx Γx ′ and assume by contradiction that d X/Γ (Γx, Γx ′ ) = 0. Then there exists a sequence of points in Γx converging to a point y in Γx ′ , and since orbits are closed, y belongs to Γx too. Therefore the two orbits coincide, which we assumed not. As a consequence, whenever Γx Γx 2.3. Synthetic Ricci curvature lower bounds. We briefly recall here the definition of RCD * spaces, and we refer to [Stu06a, Stu06b, LV09, BS10, AGS14, Gig15, AGMR15, EKS15, AMS19] for more details about synthetic curvature-dimension conditions and calculus on metric measure spaces. There are different ways to define the curvature-dimension condition, that are now known to be equivalent in the case of infinitesimally Hilbertian m.m.s. (see for example [EKS15,Theorem 7]). We chose to give here only the definitions of the CD * (K, N) condition and infinitesimally Hilbertian m.m.s., since this will be the framework of the paper. For κ, s ∈ R, we introduce the generalised sine function For (t, θ) ∈ [0, 1] × R + and κ ∈ R, the distortion coefficients are defined by if κθ 2 0 and κθ 2 < π 2 For a metric space (X, d), let P 2 (X) be the space of Borel probability measures µ over X with finite second moment, i.e. satisfying for some (and thus, for every) x 0 ∈ X. The L 2 -Wasserstein distance between µ 0 , µ 1 ∈ P 2 (X) is defined by where q is a Borel probability measure on X × X with marginals µ 0 , µ 1 . A measure q ∈ P(X 2 ) achieving the minimum in (2) is called an optimal coupling. The L 2 -Wasserstein space (P 2 (X), W 2 ) is a complete and separable space, provided (X, d) is so. Let P 2 (X, d, m) ⊂ P 2 (X) denote the subspace of m-absolutely continuous measures and P ∞ (X, d, m) the set of measures in P 2 (X, d, m) with bounded support.
Definition 2.6. Let K ∈ R and N ∈ [1, ∞). A metric measure space (X, d, m) satisfies the curvaturedimension condition CD * (K, N) if and only if for each µ 0 , µ 1 ∈ P ∞ (X, d, m), with µ i = ρ i m, i = 0, 1, there exists an optimal coupling q and a W 2 -geodesic (µ t ) t∈[0,1] ⊂ P ∞ (X, d, m) between µ 0 and µ 1 such that for all t ∈ [0, 1] and Given a metric measure space (X, d, m), the Sobolev space W 1,2 (X, d, m) is by definition the space of L 2 (X, m) functions having finite Cheeger energy, and it is endowed with the natural norm f 2 W 1,2 := f 2 L 2 + 2Ch( f ) which makes it a Banach space. Here, the Cheeger energy is given by the formula where |D f | w denotes the weak upper differential of f . The metric measure space (X, d, m) is said to be inifinitesimally Hilbertian if the Cheeger energy is a quadratic form (i.e. it satisfies the parallelogram identity) or, equivalently, if the Sobolev space W 1,2 (X, d, m) is a Hilbert space.
Definition 2.7. Let K ∈ R and N ∈ [1, ∞). We say that a metric measure space (X, d, m) is an RCD * (K, N) space if it is infinitesimally Hilbertian and it satisfies the CD * (K, N) condition.
Remark 2.7.1 (The case N = 1). If (X, d, m) is a compact RCD * (K, N) space with N = 1, then by Kitabeppu-Lakzian [KL16] we know that (X, d, m) is isomorphic either to a point, or a segment, or a circle. Hence, all the statements of this paper will hold trivially. For instance: • The revised first Betti number upper bound b 1 (X) ≤ 1 holds trivially; • The torus stability holds trivially since b 1 (X) = 1 only if (X, d, m) is isomorphic to a circle.
Without loss of generality, we will thus assume N ∈ (1, ∞) throughout the paper to avoid trivial cases.
Remark 2.7.2 (Other synthetic notions: CD(K, N), CD loc (K, N), RCD(K, N)). For K, N ∈ R, N ≥ 1 one can consider the τ-distortion coefficients Analogously to Definition 2.7, one can define the class of RCD(K, N) spaces as those CD(K, N) spaces which in addition are infinitesimally Hilbertian. It is clear from the above discussion that RCD(K, N) implies RCD * (K, N), and that RCD * (K, N) implies RCD(K * , N). An important property of RCD * (K, N) spaces is the essential non-branching [RS14], roughly stating that every W 2 -geodesic with endpoints in P 2 (X, d, m) is concentrated on a set of non-branching geodesics. This has been recently pushed to full non-branching in [Den20]. The local version of CD(K, N), called CD loc (K, N), amounts to require that every point x ∈ X admits a neighbourhood U(x) such that for each pair µ 0 , µ 1 ∈ P ∞ (X, d, m) supported in U(x) there exists a W 2geodesic from µ 0 to µ 1 (not necessarily supported in U(x)) satisfying the CD(K, N) concavity condition. For essentially non-branching spaces, it is not hard to see that CD * (K, N) is equivalent to CD loc (K, N). It is much harder to establish the equivalence in turn with CD(K, N). This was proved for essentially non-branching spaces with finite total measure in [CM]. In particular it follows that, for spaces of finite total measure, the conditions RCD loc (K, N), RCD(K, N) and RCD * (K, N) are all equivalent.
We state here some well-known properties of RCD * (K, N) spaces that we are going to use throughout the paper. First of all, we have the following natural scaling properties: The following sharp Bishop-Gromov volume comparison was proved in [Stu06b] for CD(K, N) spaces, then generalised to non-branching CD loc (K, N) spaces in [CS12], and to essentially non-branching CD loc (K, N) spaces in [CM18]. In particular it holds for RCD * (K, N) spaces. It will be useful in proving the appropriate upper bound for the revised first Betti number b 1 (X).
Remark 2.8.1. The Bishop-Gromov volume comparison implies that RCD * (K, N) spaces are locally doubling and thus proper. It is also not hard to check directly from the Definition 2.6 that supp m (and thus X, since we are assuming throughout that X = supp m) is a length space. Since a proper length space is geodesic, we have that RCD * (K, N) spaces are proper and geodesic. Thus, without loss of generality, we will assume that all the metric spaces in the paper are proper and geodesic.
2.4. Almost Splitting. We recall some results from [MN19] that we will use in the proofs, starting from an Abresh-Gromoll inequality on the excess function. For a metric measure space (X, d, m) we consider two points p, q and define the excess function as: For radii 0 < r 0 < r 1 , let A r 0 ,r 1 ({p, q}) be the annulus around p and q: We will use the following estimates, contained in [MN19, Theorem 3.7, Corollary 3.8 and Theorem 3.9].
The following integral estimate holds: (4) The almost splitting theorem for RCD * spaces states that if there exist k points in (X, d, m) that are far enough, and whose excess function and derivatives satisfy the appropriate smallness condition, then the space almost splits k Euclidean factors, meaning that (X, d, m) is mGH-close to a product R k × Y, for an appropriate RCD * metric measure space (Y, d Y , m Y ). More precisely, we follow the notation of [MN19, Theorem 5.1], where p i + p j denotes a point and d p is the distance function d p (·) = d(p, ·): Theorem 2.12. Let ε > 0, N ∈ (1, ∞) and β > 2. Then there exists δ(ε, N) > 0 with the following property. Assume that, for some δ ≤ δ(ε, N), the following holds: and for all r ∈ [1, δ −1 ]: More precisely: Theorem 2.12 was proved in [MN19] by Naber and the second named author, building on top of Gigli's proof of the Splitting Theorem for RCD * (0, N) spaces [Gig13], after Cheeger-Gromoll Splitting Theorem [CG72] and Cheeger-Colding's Almost Splitting Theorem [CC96].
2.5. Structure of RCD * (K, N) spaces and rectifiability. We collect here some known results about the structure of RCD * (K, N) spaces, which extended to the RCD * (K, N) setting previous work on Ricci limit spaces [Col97, CC97, CC00a, CC00b, CN12]. They will be used in order to prove that for ε > 0 small enough, a compact RCD * (−ε, N) space (X, d, m) with b 1 (X) = ⌊N⌋ and diam(X) = 1 is ⌊N⌋-rectifiable and the measure m is absolutely continuous with respect to the Hausdorff measure H ⌊N⌋ .
We first recall the notion of k-rectifiability for metric and metric measure spaces.
Definition 2.13 (k-Rectifiability). Let k ∈ N. A metric measure space (X, d, m) is said to be (m, k)-rectifiable as a metric space if there exists a countable collection of Borel subsets {A i } i∈I such that m(X \ i∈I A i ) = 0 and there exist bi-Lipschitz maps between A i and Borel subsets of R k . A metric measure space (X, d, m) is said to be k-rectifiable as a metric measure space if, additionally, the measure m is absolutely continuous with respect to the Hausdorff measure H k .
We next recall the definitions of tangent space and of k-regular set R k .
The set of all tangent spaces of (X, d, m) at x is denoted by Tan(X, d, m, x).
Definition 2.15. Let (X, d, m) be an RCD * (K, N) space for N ∈ (1, ∞) and K ∈ R. For any k ∈ N, the k-th regular set R k is given by the set of points x ∈ X such that tangent space at x is unique and equal to the In [MN19, Theorem 1.1] it was proved that for any RCD * (K, N) space (X, d, m), the k-regular sets R k for k = 1, . . . , ⌊N⌋ are (m, k)-rectifiable as a metric spaces and form an essential decomposition of X, i.e.
A subsequent refinement by the independent works [KM18,DPMR17,GP] showed that the measure m restricted to R k is absolutely continuous with respect to H k . Moreover, in [BS20], E. Bruè and D. Semola showed that there exists exactly one regular set R k having positive measure. It is then possible to define the essential dimension of an RCD * (K, N) space as follows.
Observe that, as a consequence, any RCD * (K, N) space of essential dimension equal to k is k-rectifiable as a metric measure space.
We finally state two theorems that will be used in the final part of the paper, to show that an RCD * (K, N) space with b 1 (X) = N ∈ N and diam(X) 2 K ≥ −ε is mGH-close and bi-Hölder homeomorphic to a flat torus T N .
Then one of the following holds.
In the following statement, we rephrase Theorem 1.10 of [KM21]: Theorem 2.18 ([KM21, Theorem 1.10]). Let (M, g) be a compact manifold of dimension N (without boudary). There exists ε = ε(M) > 0 such that the following holds. If (X, d, m, x) is a pointed RCD * (K, N) space for some K ∈ R satisfying d pmGH (X, M) < ε, then m = c X H N for some c X > 0 and (X, d) is bi-Hölder homeomorphic to M.
2.6. Covering spaces, universal cover and revised fundamental group. We first discuss the definition of covering spaces, universal cover, revised fundamental group, and actions of groups of homeomorphisms over topological spaces. Then we focus on length metric measure spaces and see that the RCD * (K, N) condition can be lifted to the total space of an RCD * (K, N) base, when having a covering map.
2.6.1. Covering spaces. Let us provide some definitions and results related to coverings spaces from [Spa66,Hat02]. In particular, we state the notion of a group acting properly discontinuously as it appears in these references. Note that sometimes this is defined differently.
We say that a topological space Y is a covering space for a topological space X if there exists a continuous map p Y,X : Y → X, called covering map, with the property that for every point x ∈ X there exists a neighbourhood U ⊂ X of x such that p −1 Y,X (U) is the disjoint union of open subsets of Y and so that the restriction of p Y,X to each of these subsets is homeomorphic to U. By definition, the covering map is a local homeomorphism. Two covering spaces Y, Y ′ of X are said to be equivalent if there exists an homeomorphism between them, h : If X is path-connected then the cardinality of p −1 Y,X (x) does not depend on x ∈ X. We recall that given a topological space Z and z ∈ Z, the fundamental group of Z, π 1 (Z, z), is the group of the equivalence classes under based homotopy of the set of closed curves from [0, 1] to Z with endpoints equal to z. Any covering map p Y,X induces a monomorphism p Y,X♯ : π 1 (Y, y 0 ) → π 1 (X, p Y,X (y 0 )); moreover, when both Y and X are path-connected, the cardinality of Before defining the group of deck transformations of a covering space, we introduce some terminology of group actions.
Definition 2.19. A group of homeomorphisms G of a topological space Y is said to act effectively or faithfully if y∈Y g | g(y) = y = {e}, where e denotes the identity element of G. It acts without fixed points or freely if the only element of G that fixes some point of Y is the identity element. We say that G acts discontinuously if the orbits of G in Y are discrete subsets of Y and we say that G acts properly discontinuously if every y ∈ Y has a neighbourhood U ⊂ Y so that U ∩ gU = ∅ for all g ∈ G \ {e} 1 .
So, acting properly discontinuously implies acting discontinuously and without fixed points, and every free action is effective.
The group of deck transformations of a covering space Y of X is the group of self-equivalences of Y: By the unique lifting property, G(Y | X) acts without fixed points. Combining this fact with the definition of covering map, we see that G(Y | X) also acts properly discontinuously on Y.
If Y is connected and locally path-connected, then p Y,X is regular if and only if the group G(Y | X) acts transitively on each fibre of p Y,X . In this case, for any y 0 ∈ Y we have: • An isomorphism of groups: • A bijection between any fibre of p Y,X and G(Y | X); • A homeomorphism of spaces: Definition 2.20 (Universal cover of a connected space). Given a connected topological space X, a universal covering space X for X is a connected covering space for X such that for any other connected covering space Y of X there exists a map f : X → Y that forms a commutative triangle with the corresponding covering maps, i.e. p Y,X • f = p X,X .
Since we do not require X to be semi-locally simply connected, then X might not be simply connected. Thus, the group G( X | X) of deck transformations of X might not be isomorphic to the fundamental group of X. However, G( X | X) acts properly discontinuously on X, transitively on each fibre of p Y,X ; thus p X,X is regular. Moreover, any (connected) covering space of X is covered by X. In particular, universal covering spaces of a connected and locally path-connected space are equivalent.
Recall also that for a connected topological space Y and a group G of homeomorphisms of Y acting properly discontinuously on Y, the projection map Y → Y/G is a regular covering whose group of deck transformations coincides with G, i.e. G(Y | Y/G) = G. We conclude this subsection summarising some results that will be used later. 2.6.2. Coverings of metric spaces and RCD * (K, N) spaces. We now discuss some definitions and results related to coverings of metric spaces. For more details we refer to [SW04a] and [MW19]. Let (X, d X ) be a length metric space and p Y,X : Y → X be a covering map. The length and metric structure of X can be lifted to Y so that the covering map becomes a local isometry. Explicitly, denoting by L X the length structure of X, define the metric This lifting process implies that Y is complete whenever X is so. In particular, if X is compact, then Y will be a complete, locally compact length space, thus proper [BBI01, Proposition 2.5.22].
If X is locally compact and m X is a Borel measure on it, we can lift m X to a Borel measure m Y on Y that is locally isomorphic to m X . In order to define m Y , denote by B(Y) the family of Borel subsets of Y and consider the following collection of subsets of Y: Note that Σ is stable under intersections and that Y is locally compact given that p Y,X is a local isometry. Thus, the smallest σ-algebra that contains Σ equals B(Y). For E ∈ Σ, define m Y (E) := m X p Y,X (E) and then extend it to all B(Y).
From now on, all the covering spaces will be endowed with this metric and measure. The following result was proved in [MW19].
Theorem 2.22. For any K ∈ R and any N ∈ (1, ∞), any RCD * (K, N) space admits a universal cover space ( X, d X , m X ) which is itself an RCD * (K, N) space.
We now state Sormani-Wei's definition of revised fundamental group [SW04a].
Definition 2.23. (Revised fundamental group) Given a complete length metric space (X, d X ) that admits a universal cover ( X, d X ), the revised fundamental group of X, denoted byπ 1 (X), is defined to be the group of deck transformations G( X | X).
Recall that the covering map p X,X associated to the universal cover space of X is regular and thusπ 1 (X) acts transitively on each fibre of p X,X and properly discontinuously on X by homeomorphisms; such homeomorphisms are measure-preserving isometries on X, provided X is endowed with the lifted distance and measure of X, as described above.
We conclude this subsection by mentioning two properties that will be used later. First, for a covering map p Y,X : Y → X, one can prove (by lifting geodesics of X to Y) that for any x, x ′ ∈ X and y ∈ Y with . It follows that if p Y,X is regular, and thus G(Y | X) acts transitively on its fibres, then for any y, y ′′ The second property is that a quotient space Y/H as in Proposition 2.21 is an RCD * (K, N) space provided either X or Y is an RCD * (K, N) space. We give more details below. In Theorem 1.1 we will use this fact to get an upper bound on the revised first Betti number of an RCD * (K, N) by passing to a quotient space ( X/H for H = [π 1 (X),π 1 (X)]); this fact will be also useful in Lemma 6.3 to infer that the GH convergence of a sequence of quotient spaces can be promoted to mGH convergence. Assume that (X, d X , m X ) is an RCD * (K, N) space. Then (X, d X ) is complete, separable, proper and geodesic. Since p Y,X is a regular covering map, we can apply [BBI01, Proposition 3.4.16] stating that the length metrics on X are in 1-1 correspondence with the G(Y | X)-invariant length metrics on Y; thus (Y, d Y ) is a length metric space. Since p Y,X is a local isometry, we automatically get that (Y, d Y ) is a complete and locally compact space. Moreover, by our assumption on p −1 Y,X (x), (Y, d Y ) is separable. Now every complete locally compact length space is geodesic [BBI01, Theorem 2.5.23]. Hence, (Y, d Y ) is a complete, separable and geodesic space.
In order to prove that N) if and only if it is infinitesimally Hilbertian and it satisfies the strong CD e (K, N) condition, defined as in Definition 3.1 of [EKS15]. Since (X, d X , m X ) is an RCD * (K, N) space, by [EKS15, Theorem 3.17, Remark 3.18] we infer that (X, d X , m X ) satisfies the strong CD e (K, N) condition. Now [EKS15, Theorem 3.14] says that on a geodesic m.m.s. the strong CD e (K, N) condition is equivalent to the strong local CD e loc (K, N) condition, thus in particular (X, d X , m X ) satisfies the strong local CD e loc (K, N) condition. Now each point y ∈ Y has a compact neighbourhood U y such that (U y , d Y | U y ×U y , m Y U y ) is isomorphic as metric measure space to (p Y,X (U y ), d X | p Y,X (U y )×p Y,X (U y ) , m X p Y,X (U y ) ). It follows that the strong local CD e loc (K, N) condition satisfied by (X, d X , m X ) passes to the covering (Y, d Y , m Y ). Since Y is geodesic, then by [EKS15, Theorem 3.14] it also satisfies the strong CD e (K, N) condition.
It remains to show that (Y, d Y , m Y ) is infinitesimally Hilbertian. This follows by a partition of unity on Y made by Lipschitz functions with compact support contained in small metric balls isomorphic to metric balls in X, using the fact that the Cheeger energy is a local object (see [AGS14,Gig15]). Indeed, the validity of the parallelogram identity for the Cheeger energy on Y can be checked locally (on each small ball) using a partition of unity. Since such small balls in Y are isomorphic to small balls of X where the Cheeger energy satisfies the parallelogram identity, we conclude that the Cheeger energy on Y satisfies the parallelogram identity as well.
Thus (Y, d Y , m Y ) is infinitesimally Hilbertian, satisfies the CD e (K, N) condition and supp(m Y ) = Y. It follows by [EKS15,Theorem 3.17 The converse implication can be proved with analogous arguments.
Proof. By [SW04a, Proposition 6.4 and Lemma 6.2] and Bishop-Gromov volume comparison theorem, for any compact RCD * (K, N) space (X, d, m), its revised fundamental groupπ 1 (X) can be generated by a set of cardinality at most N(δ 0 , diam(X)) < ∞, where δ 0 corresponds to the δ 0 -cover of X so that X = X δ 0 and N(δ 0 , diam(X)) is the maximal number of balls in X of radius δ 0 in a ball of radius diam(X).
Corollary 2.26. Let (X, d, m) be a compact RCD * (K, N) space, for some K ∈ R and N ∈ (1, ∞). Then for any normal subgroup H of the revised fundamental groupπ 1 (X), the metric measure space ( X/H, d X/H , m X/H ) is an RCD * (K, N) space which is covered by X and covers X.
Proof. Since from Proposition 2.25 we know thatπ 1 (X) is finitely generated, then it is at most countable. Thus the cardinality of each fibre of the covering map is at most countable. We can thus conclude using Proposition 2.21 and Lemma 2.24.
Remark 2.26.1. Since the group of deck transformations G(Y | X) acts properly discontinuously on Y, the semi-metric d Y/H (Hy, defined on the quotient space Y/H = {Hy | y ∈ Y} is actually a metric (this can be seen, for example, using Section 2.2). We also observe that, under the assumptions of Proposition 2.21, since Y/H is a cover of X it can also be endowed with the lifted metric of X as defined in (6). We note that this metric coincides with (8), so we will use the quotient metric of Y/H whenever it is convenient. Notice that, in particular, all the covering maps appearing in Proposition 2.21 are local isometries.

UPPER BOUND ON THE REVISED FIRST BETTI NUMBER: b 1 ≤ ⌊N⌋
In this section we obtain an upper bound for the revised first Betti number of an RCD * (K, N) space with K ≤ 0 and N ∈ (1, ∞). In the case of smooth manifolds, the estimate is due to M. Gromov [Gro81] and S. Gallot [Gal83] (compare also [Pet16, Section 9.2]).
We consider a compact geodesic space admitting a universal cover and define its revised first Betti number as the rank of the abelianisation of the revised fundamental group, whenever the abelianisation is finitely generated. Indeed, the fundamental theorem of finitely generated abelian groups states that for any finitely generated abelian group G there exist a rank s ∈ N, prime numbers p i and integers s i such that G is isomorphic to Z s × Z s 1 p 1 × · · · × Z s l p l .
Definition 3.1. Let (X, d) be a compact geodesic space admitting a universal cover. Letπ 1 (X) be its revised fundamental group, set H := [π 1 (X),π 1 (X)] the commutator and Γ :=π 1 (X)/H. Then we define the revised first Betti number of X as b 1 (X) := From now on, we denote byX the quotient space X/H: By Proposition 2.21, we know thatX is a cover of X; moreover, Γ acts onX as an abelian group of isometries. Since the action ofπ 1 (X) is properly discontinuous, the same is true for Γ. In particular, the action is discontinuous and all the orbits Γx, x ∈ X, are discrete. The first step in proving the upper bound on the revised first Betti number consists in showing the appropriate analog of a result of Gromov [Gro07,Lemma 5.19]. In the case of smooth manifolds, compare with [Pet16, Lemma 2.1, Section 9.2] for k = 1 and for general k ∈ N with [Col97, Lemma 3.1].
Proof. We first find a subgroup Γ ′′ ≤ Γ of finite index and generated by elements that satisfy (11) for k = 1. For any ε > 0, set r ε = 2 diam(X) + ε and let Γ ε be the subgroup of Γ generated by the set Observe that the previous set is not empty since, because of (7), there exists γ ∈ Γ such that dX(γ(x), x) ≤ diam(X). EndowX/Γ ε with the quotient topology and the distance dX /Γ ε induced by dX, c.f. (1). Let π ε :X →X/Γ ε be the covering map.
Remark 3.2.1. With a more careful analysis, a similar version of Lemma 3.2 holds true if one replaces geodesic space by length spaces. c.f. [Gro07]. Furthermore, the same conclusion holds if we considerX/T instead ofX, where T is any torsion subgroup of Γ.
Remark 3.2.2. Note thatX is not compact. Indeed, for anyx ∈X and corresponding Γ ′ given by Lemma 3.2, the orbit Γ ′x = {γ(x), | γ ∈ Γ ′ } is countable, since Γ ′ acts properly discontinuously onX and Γ ′ is isomorphic to Z b 1 . Now, if by contradictionX is compact, then Γ ′x has a converging subsequence {γ i (x)}. By using either that the action is properly discontinuous or property (10), it is not difficult to show that {γ i (x)} must be a constant sequence starting from i large enough, giving a contradiction.
Remark 3.2.3. It is not difficult to see that Γ ′ is a closed discrete group in the compact-open topology. Recall if a sequence of isometries γ i in Γ ′ converges to γ in the compact-open topology, then it converges uniformly on every compact subset and in particular for anyx ∈X we have γ i (x) → γ(x). We know that for any fixed x ∈X, the only converging sequences in the orbit Γ ′x are (definitely) constant sequences. Thus there exist γ ∈ Γ ′ and i 0 ∈ N such that for all i ≥ i 0 we have γ i (x) = γ(x). Therefore any converging sequence γ i in Γ ′ is constantly equal to an element γ of Γ ′ , yielding that Γ ′ is closed and discrete.
In the following, we consider a compact RCD * (K, N) space, (X, d, m) with N ∈ (1, ∞). By Theorem 2.22, we know that it admits a universal cover space, ( X, d, m), that satisfies the RCD * (K, N) condition. Using the same notation as in Lemma 3.2, by Corollary 2.26, the quotient m.m.s. (X,d,m) is also an RCD * (K, N) space. Since by Proposition 2.25 we know that the revised fundamental groupπ 1 (X) is finitely generated, we infer that the revised first Betti number of (X, d, m) is finite.
We are now ready to prove the first main result of the paper, namely the desired upper bound for b 1 (X). This is done by combining Lemma 3.2 with Theorem 2.8 for (X,d,m), and generalises to the non-smooth RCD setting the celebrated upper bound proved in the smooth setting by M. Gromov [Gro81] and S. Gallot [Gal83] (see also [Pet16, Theorem 2.2, Section 9.2] and [Gro07, Theorem 5.21]).
By the definition of I r , |I r | is non decreasing with respect to r. If r = 1 then {γ i } b 1 i=1 ⊂ I r and thus b 1 ≤ |I r | for r ≥ 1.
(15) For arbitrary r ∈ N, it is easy to check that Now we apply the relative volume comparison theorem, Theorem 2.8, to obtain an upper bound on the cardinality of I r . Since the right hand sides of both equations in Theorem 2.8 are non increasing as a function of K, we can assume that K < 0. Thus, That is, C r (N, ·) : [0, −KD 2 /(N − 1)] → R is the function given by By (15) and since C r (N, t) is non decreasing as a function of t, we have b 1 (X) ≤ C r (N, −KD 2 /(N − 1)). By using the Taylor expansion of sinh, we calculate that Thus, for small t we have Now assume by contradiction that there exists a sequence ε i ↓ 0 and RCD * (K i , N) metric measure spaces Thanks to (16) and (17), we know that for any integer r ≥ 1 we have (N − 1)).
For ε i small enough, we can apply (18), so that for all r ∈ N, r ≥ 1 Thus for r large enough This concludes the proof of the second assertion.
In order to prove the first assertion, set and observe that, thanks to (17), C(N, t) is bounded by C r (N, t/(N − 1)). Since it is a bounded supremum of integer numbers, C (N, t) is an integer. Moreover, the flat torus T ⌊N⌋ is an RCD * (0, N) space with b 1 (T N ) = ⌊N⌋, hence C(N, t) ≥ ⌊N⌋. The previous argument also shows that for t ≤ ε(N), b 1 (X) ≤ N, thus for any t ≤ ε(N) we have ⌊N⌋ ≤ C(N, t) ≤ N. This implies that C(N, t) = ⌊N⌋ for any t ≤ ε(N). As a consequence, C(N, t) is the desired function tending to ⌊N⌋ as t → 0.
Remark 3.2.4. In the case of n-dimensional manifolds, Gallot proved an optimal bound for b 1 (M) and expressed the function C(n, t) as ξ(n, t)n, where ξ(n, t) is an explicit function tending to one as t tends to zero [Gal83, Section 3].

Proof. By Lemma 3.2 there exists a subgroup of deck transformations
Arguing as in the proof of Theorem 1.1 we get that, for any r ∈ N, the number of disjoint balls of radius k/2 in B k/2+2kr (x) is larger than or equal to the number of elements in The cardinality of I r equals (2r + 1) b 1 (X) . Then for R > 1 write ⌊R⌋ = k(2r + 1) and get that B R (x) contains at least (2r + 1) b 1 (X) = ⌊R/k⌋ b 1 (X) disjoint balls of radius k/2.

CONSTRUCTION OF MGH-APPROXIMATIONS IN THE EUCLIDEAN SPACE
This section is devoted to proving Theorem 4.1, which corresponds to the non-smooth RCD version of [Col97, Lemma 3.5]. The main goal is to show that if (X, d, m) is an RCD * (−δ, N) space with δ = δ(ε, N) small enough, diam(X) = 1 and b 1 (X) = ⌊N⌋, then the covering space (X, dX, mX) (defined in (9)) is locally (on suitably large metric balls) mGH close to the Euclidean space R ⌊N⌋ .
The proof consists in applying inductively the almost splitting theorem. More precisely, we show that (X, dX, mX) is locally (on suitably large metric balls) mGH close to a product R k × Y k by induction on k = 1, . . . , ⌊N⌋. Since the diameter of the covering spaceX is infinite (see Remark 3.2.2), the base case of induction k = 1 will follow by carefully applying the almost splitting theorem. As for the inductive step, thanks to Corollary 3.3 we will prove a diameter estimate on Y k that allows us to apply the almost splitting theorem on Y k . We will conclude by deducing the almost splitting of an additional Euclidean factor by constructing an ε-mGH approximation into R k+1 × Y k+1 .
Step 1: Estimate on the sup of the excess. We can apply Theorem 2.11 and infer that there existr =r(N) > 0, C = C(N) > 0, α = α(N) ∈ (0, 1) such that the estimate (4) centred at x ε holds for d δ . By scaling back to the metric d, such an estimate can be written as follows: sup We aim to choose δ > 0 such that (19) can be turned into the following: Hence we first require so that (19) applies to all radii r ∈ [1, δ −1 ]. Secondly, we need C(N) δ α β (δ −1 ) 1+α < δ/2, so the right-hand side of (19) is bounded above by δ/2. That means Such a choice is possible since the assumption β > (2 + α)/α ensures that the exponent on the left-hand side is strictly positive. By choosing δ > 0 sufficiently small so that both conditions (a) and (b) are satisfied, we obtain from (19) that estimate (20) holds.
As in step 1, we aim to choose δ > 0 so that (21) implies the following: Hence we require In order for the right-hand side of (21) to be less than or equal to δ/2 we need Note that since β > 2, the exponent on the left-hand side is strictly positive.
Proposition 4.2 can be in particular applied to the covering space (X, dX, mX). Indeed, thanks to Corollary 2.26 it is an RCD * space and it is not compact (thus it must have infinite diameter, since it is proper), as it was pointed out in Remark 3.2.2. This gives the base case of induction, k = 1.
Observe that for the base case of induction (i.e. in Corollary 4.3), we did not use the assumptions on the diameter and revised first Betti number. These assumptions will play a key role in the following, instead. Let us state the induction hypothesis.
Assumption A k : Fix N ∈ (1, ∞) and let k ∈ N with k < ⌊N⌋. For all η ∈ (0, 1) there exists δ k = δ k (η, N) > 0 such that for all δ ∈ (0, δ k ], the following holds: if (X, d, m) and (X, dX, mX) are as in Theorem 4.1, then there existsx k,η ∈X and a pointed RCD In order to prove A k+1 given A k we aim to apply Proposition 4.2 to the space Y k,η : in this way, Y k,η will almost split a line, thusX will almost split an additional Euclidean factor, yielding A k+1 . To this aim, the following diameter estimate will be key.
Lemma 4.4. Assume that A k holds. For any η ∈ (0, 1), let δ k (η, N) > 0, (X, d, m) and (X, dX, mX) be as in A k and (Y k,η , d Y k,η , m Y k,η , y k,η ) be the corresponding RCD * (0, N − k) p.m.m.s.. Then there exist c N ∈ (0, 1) and η 0 (N) > 0 such that for all η ∈ (0, η 0 (N)] and for all δ ∈ (0, δ k (η, N)], it holds: Proof. We argue by contradiction. Assume there exist a sequence η i ↓ 0, corresponding Let i be sufficiently large so that η i < 1. By Corollary 3.3 we know that BX i η −1 i (x i ) contains at least (⌊η −1 i ⌋) b disjoint balls of radius 1/2, at positive mutual distance. Using (23) we infer that, for i large enough, the ball B R k ×Y i η −1 i ((0 k , y i )) in R k × Y i also contains at least (⌊η −1 i ⌋) b disjoint balls of radius 1/2. Rescale the metric of R k × Y i by a factor η i and denote the resulting space as (R k × Y i ) η i . Then for large enough i the ball of radius 1 in (R k × Y i ) η i centred at (0 k , y i ) contains at least ( ⌊η −1 i ⌋) b disjoint balls of radius η i 2 at positive mutual distance. Furthermore, since η i diam(B Y i η −1 i (y i )) tends to zero as i tends to infinity, when taking the Gromov-Hausdorff limit of such balls we obtain: As a consequence, for large enough i, B (R k ) η i 1 (0) contains at least (⌊η −1 i ⌋) b disjoint balls of radius η i 2 . Denote by ω k the volume of B R k 1 (0). Since we only rescaled the metric of R k × Y i by a factor η i , the mass of equals ω k (η −1 i ) k and the mass of a ball of radius η i 2 in B (R k ) η i 1 (0) equals ω k (1/2) k . Hence, However, since 1 ≤ k < ⌊N⌋ = b and η i → 0, the estimate (24) cannot hold for i sufficiently large.
Remark 4.4.1. Notice that we used the hypotheses diam(X) = 1 and b := b 1 (X) = ⌊N⌋ to have a given number of disjoint balls of radius 1/2 in a ball inX of radius larger than 1.
We next combine Lemma 4.4 and Proposition 4.2 in order to prove that the space Y k,η almost splits a line, for η > 0 small enough depending on ε > 0.
Proposition 4.5. Assume that A k is satisfied. Then for any ε ∈ (0, 1) there exists η(ε, N) > 0 such that the following holds. For any η ∈ (0, η(ε, where δ 1 (ε, N) > 0 is the quantity given by Proposition 4.2 and c N , η 0 (N) are defined in Lemma 4.4. Then by assumption A k and Lemma 4.4, for any η ∈ (0, η(ε, N)] and for all δ ∈ (0, δ k (η, N)], if (X, d, m) is an RCD * (−δ 2β , N) space as in assumption A k , then there existx k,η ∈X and a pointed RCD Let ξ > 0 be such that c N η −1 = 2ξ −β . Our choice of η(ε, N) ensures that for any η ∈ (0, η(ε, N)] we have ξ ∈ (0, δ 1 (ε, N)]. Therefore we can apply Proposition 4.2 to Y k,η and get that there exist y ∈ Y k,η and a pointed RCD It remains to show that y ∈ B Y k,η η −1 /2 (y k,η ) and that B . From the proof of Proposition 4.2, we know that y is a midpoint of a geodesic between two points p, q at distance equal to c N η −1 . Since Y k,η is a geodesic space and c N ∈ (0, 1), it is easily seen that (25) implies that there exists a point q ∈ B Y k,η η −1 (y k,η ) such that d Y k,η (q, y k,η ) = c N η −1 . Then, in the proof of Proposition 4.2 we can chose p = y k,η , q ∈ Y k,η with d Y k,η (q, y k,η ) = c N η −1 and y a midpoint of a geodesic between p and q.
We are now in position to prove Theorem 4.1.
Therefore, there existsx k+1,η 1 in BX η −1 1 (x k,η 1 ) such that We aim to show that We first claim that Indeed, since φ is an η 1 -mGH approximation, by the definition ofx k+1,η 1 and y we have The claim (27) follows by triangle inequality.
Thanks to Proposition 7.2, there exists a (C 2 ε 1 )-mGH approximation: Since the ball centred at (0 k , y) of radius ε −1 ≤ ε −1 1 / √ 2 is included in the previous product of balls, we can use again Proposition 7.1 to construct a (C 1 C 2 ε 1 )-mGH approximation out of ϕ: The composition of ϕ 1 with φ 1 then gives a (2C 1 η 1 + C 1 C 2 ε 1 )-mGH approximation: Thanks to our choices of C, ε 1 and η 1 , the map f is an ε-mGH approximation and the claim (26) is proved.

PROOF OF THEOREM 1.2, FIRST CLAIM
In this section we prove the first part of the main Theorem 1.2, by combining Theorem 4.1 with the structure theory of RCD * (K, N) spaces [MN19, KM18, GP, BS20]. More precisely we show the following result, which in turn immediately implies the first claim of Theorem 1.2 by a standard scaling argument.
In [MN19, Theorem 6.8], the authors proved that for any ε > 0 there exists δ > 0 such that if (X, d, m) is an RCD * (−δ, N) space and a ball of radius δ −1 is δ-mGH close to a Euclidean ball of the same radius in R ⌊N⌋ , then there exists a subset (of large measure) U ε of the unit ball which is (1 + ε) bi-Lipschitz to a subset of R ⌊N⌋ . In order to construct U ε and the bi-Lipschitz map into R ⌊N⌋ , they showed the existence of a function u on the unit ball which, restricted to any ball B s (x) centred at a point x of U ε , is an (εs)-mGH isometry. We summarise these results in the following statement.
Corollary 5.3. For every N ∈ (1, ∞) there exists δ 0 = δ 0 (N) > 0 with the following property. Let (X, d, m) be an RCD * (−δ, N) space for some δ ∈ [0, δ 0 ) and assume that for some x 0 ∈ X it holds Then the essential dimension of X is equal to ⌊N⌋ and (X, d, m) is ⌊N⌋-rectifiable as a metric measure space.
Proof. By the definition of the dimension of RCD spaces, we know that there exists a unique n ∈ N, with n ≤ ⌊N⌋, such that the n-th regular stratum R n has positive measure. Therefore, by definition of R n for m-a.e. x ∈ X, tangent cones at x are unique and equal to the Euclidean space (R n , d R n , L n ). Now assume by contradiction that n < ⌊N⌋. Because of Theorem 5.2, (28) implies the existence of a set U ε satisfying properties 1 to 3, with m(U ε ) > 0. As a consequence, there exists x ∈ U ε with unique tangent cone equal to R n . Property 3 then implies that the unit ball in R n is ε-GH close to the unit ball in R ⌊N⌋ , which is impossible for n < ⌊N⌋ and ε > 0 sufficiently small. Therefore, ⌊N⌋ is the essential dimension of (X, d, m) and (X, d, m) is ⌊N⌋-rectifiable as a metric measure space.
The combination of Corollary 5.3 and Theorem 4.1 yields the following result.
We are now in position to prove Theorem 5.1.
Proof of Theorem 5.1. Letp :X → X be the covering map and denote by R ⌊N⌋ (X) the ⌊N⌋-th regular set of X. Recall that mX(X \ R ⌊N⌋ (X)) = 0. Let BX r (x) be a sufficiently small ball inX such that is an isomorphism of metric measure spaces. Since for mX-a.e.x ′ ∈ BX r (x) the tangent cone is unique and equal to R ⌊N⌋ , the same is true for m-a.e. x ′ ∈ B X r (p(x)) and thus the regular set R ⌊N⌋ of X has positive m-measure. Therefore (X, d, m) has essential dimension equal to ⌊N⌋ and it is ⌊N⌋-rectifiable as a metric measure space. In particular, m ≪ H ⌊N⌋ .
If N is an integer, then m ≪ H N and (X, d, m) is a compact weakly non-collapsed RCD * (−δ, N) space. Corollary 1.3 in [Hon20] ensures that for any compact weakly non-collapsed RCD * (−δ, N) space, there exists c > 0 such that m = cH N , thus concluding the proof.
6. PROOF OF THEOREM 1.2, SECOND AND THIRD CLAIMS Now we are in position to conclude the proof of Theorem 1.2. Given a sequence of RCD * (−K i , N) spaces (X i , d i , m i ) with K i < 0 tending to zero, diam(X i ) = 1 and b 1 (X i ) = ⌊N⌋, the proof consists in applying the results of equivariant pointed Gromov-Hausdorff convergence as in Section 2.2 to the sequence (X i , d i ,x i ) and subgroups Γ ′ i as in Lemma 3.2, in order to obtain equivariant convergence (up to a subsequence) to . Then we will conclude that the quotientsX i /Γ ′ i mGH converge to a flat torus, which, by applying Theorem 2.18, will imply that for large i the quotients are bi-Hölder homeomorphic to this torus. In the last step we show thatX i /Γ ′ i = X i . We start with the following lemma.
eq be a sequence of spaces that converge in equivariant pGH sense to (X ∞ , d ∞ , x ∞ , Γ ∞ ) ∈ M p eq . Assume Γ i is an abelian group, for each i ∈ N. Then Γ ∞ is an abelian group as well.
Proof. Given arbitrary γ ∞1 , γ ∞2 ∈ Γ ∞ , we will show that they commute. For that, by hypothesis there exist ε i -equivariant pGH approximations ( f i , φ i , ψ i ): satisfying the conditions of Definition 2.2 and so that ε i → 0. Take an arbitrary point z ∞ ∈ X ∞ . By the triangle inequality and for i large enough such that Applying (4) of Definition 2.2 and that φ i (γ ∞1 ) is an isometry, we see that each term in the right hand side of the previous inequality is bounded above by ε i . We conclude that The same estimate holds reversing the roles of γ ∞1 and γ ∞2 , that is: By the triangle inequality and using that Γ i is abelian, so that φ i (γ ∞2 )φ i (γ ∞1 ) = φ i (γ ∞1 )φ i (γ ∞2 ), we get: From (3) of Definition 2.2, we also have: Therefore, when taking the limit as i → ∞ we obtain d ∞ (γ ∞1 γ ∞2 z ∞ , γ ∞2 γ ∞1 z ∞ ) = 0. Since z ∞ ∈ X ∞ is an arbitrary point, we conclude that γ ∞1 and γ ∞2 commute.
We are now ready to prove the key result of this section, which directly gives the second claim of Theorem 1.2 by a standard compactness/contradiction argument.
Proposition 6.2. Let N ∈ (1, ∞) and let (X i , d i , m i ) be a sequence of RCD * (−K i , N) spaces with b 1 (X i ) = ⌊N⌋, diam(X i ) = 1 and K i > 0 such that K i ↓ 0. Fix somex i ∈X i and let Γ ′ i be as in Lemma 3.2, for k = 3. Then any Gromov-Hausdorff limit of X ′ i =X i /Γ ′ i is isometric to an ⌊N⌋-dimensional flat torus.
Remark 6.2.1. In Proposition 6.2 we require diam(X i ) = 1 instead of the bound K i diam(X i ) 2 ↓ 0. To show that the latter condition is not enough, consider a sequence X i of manifolds with K i = i and diam(X i ) = i −1 . Then K i diam(X i ) 2 ↓ 0 but any GH limit of this sequence collapses due to diam(X i ) → 0. We could also consider manifolds X i with K i = i −3 and diam(X i ) = i then K i diam(X i ) 2 ↓ 0 and any GH converging subsequence has a limit space with infinite diameter. Hence, it is necessary to have two sided uniform bounds on diam(X i ) and for simplicity we set them equal to 1.
Proof of Proposition 6.2. Set b := ⌊N⌋ = b 1 (X i ). For simplicity of notation, we will not relabel subsequences. By Theorem 4.1 and Remark 4.1.1, the sequence (X i , dX i ,x i ) converges in pointed Gromov-Hausdorff sense to (R b , d R b , 0 b ). By Gromov's compactness Theorem and stability of the RCD * (0, N) condition, there exists an RCD * (0, N) space (X, d X , m X ) with diam(X) = 1 such that X i → X in mGH sense, up to a subsequence. From Remark 3.2.3 we know that, for any i ∈ N, the groups Γ ′ i given by Lemma 3.2 are closed. Thus, by Theorem 2.4 there exist a group of isometries of R b , Γ ′ ∞ , and a subsequence (X i , dX i ,x i , Γ ′ i ) that converges in the equivariant pointed Gromov-Hausdorff sense to (R b , d R b , 0 b , Γ ′ ∞ ). Moreover, R b is the universal cover of X, and Γ ′ ∞ is contained in the corresponding group of deck transformations. We will show that R b /Γ ′ ∞ is a flat torus. To this aim, we prove that Γ ′ ∞ is isomorphic to Z b .
Let ( f i , φ i , ψ i ) be equivariant ε i -pGH approximations, ε i → 0, as in Definition 2.2: To prove (29), we first show that the claim holds for all non trivial γ i ∈ Γ ′ i and all y i ∈X i , i ∈ N. Then a convergence argument will show that the claim holds.
Step 2. We show that Γ ′ ∞ Z b . From Lemma 3.2 we know that Γ ′ i Z b . Let {γ i j } b j=1 be a set of generators for Γ ′ i . By the Arzelá-Ascoli theorem there exist a subsequence (X i k , dX i k ,x i k , Γ ′ i k ) and corresponding subsequences of isometries {γ i k 1 } ∞ k=1 , . . . , {γ i k b } ∞ k=1 that converge to γ ∞1 , . . . , γ ∞b ∈ Γ ′ ∞ , respectively. We are going to show that {γ ∞ j } b j=1 are independent generators of Γ ′ ∞ and that they have infinite order. To simplify notation consider that the whole sequence converges. Given γ ∞ ∈ Γ ′ ∞ , notice that φ i (γ ∞ ) → γ ∞ in Arzelá-Ascoli sense. Indeed, for all z ∈ R b and z i ∈X i such that dX i ( f i (z), z i ) → 0, by using the triangle inequality and (4) in Definition 2.2, and since φ i (γ ∞ ) is an isometry, we have Moreover, for any γ ∞ ∈ Γ ′ ∞ , there exist s 1 , . . . s b ∈ Z such that φ i (γ ∞ ) = γ s 1 i1 · · · γ s b ib . Then we know that the left hand side of the previous equation converges to γ ∞ , while the right hand side converges to γ s 1 ∞1 · · · γ s b ∞b . Thus, any γ ∞ ∈ Γ ′ ∞ can be written as a composition of elements in {γ ∞ j } b j=1 . We next show that {γ ∞ j } b j=1 are independent and have infinite order. Let (s 1 , . . . , s b ) ∈ Z b \ {(0, . . . , 0)}. We claim that γ s 1 ∞1 · · · γ s b ∞b id. From the previous arguments, we know that γ s 1 i1 · · · γ s b ib → γ s 1 ∞1 · · · γ s b ∞b as i → ∞. Since {γ i j } b j=1 are independent generators of Γ ′ i Z b , we have that γ s 1 i1 · · · γ s b ib id. Hence, from (30) it follows that 1 < dX i (γ s 1 i1 · · · γ s b ib f i (z), f i (z)) → d R b (γ s 1 ∞1 · · · γ s b ∞b z, z), for all z ∈ R b , and thus γ s 1 ∞1 · · · γ s b ∞b id. In conclusion, by the fundamental theorem of finitely generated abelian groups, we infer that Γ ′ ∞ Z b . Thus, R b /Γ ′ ∞ is a b-dimensional flat torus. The proposition follows now by Theorem 2.5. Proof. Suppose by contradiction that there is no such ε(N) > 0. Then there exists a sequence of compact RCD * (K, N) spaces (X i , d i , H N ) with K i diam 2 (X i ) > −ε i , b 1 (X i ) = N, ε i → 0 such that none of the X i is bi-Hölder homeomorphic to a flat torus of dimension N. Consider the rescaled spaces (X . Clearly X ′ i has diameter equal to 1 and it is an RCD * (K i diam 2 (X i , d i ), N) space with b 1 (X ′ i ) = N. Thus we can apply Proposition 6.2 and infer that any GH-limit is a flat torus T N . Moreover, from Theorem 2.17 (i) we have that (X ′ i , d ′ i , H N ) converges in mGH sense to (T N , d T N , H N ). For i large enough so that d mGH (X ′ i , T N ) ≤ ε(T N ), we can apply Theorem 2.18 and get that X ′ i is bi-Hölder homeomorphic to T N . When scaling back to the original metric, the same conclusion holds. This is a contradiction.
We can now conclude the proof of the main theorem.
Proof of the third claim of Theorem 1.2, i.e. when N ∈ N. If N = 1, the claim holds trivially (see Remark 2.7.1); thus, we can assume N ≥ 2 without loss of generality. From Corollary 6.3, we know that (X, dX) is locally (on arbitrarily large compact subsets) bi-Hölder homeomorphic to R N (thus in particular it has the integral homology of a point) and mX is a constant multiple of the N-dimensional Hausdorff measure H N . By construction, we also know that the abelianised revised fundamental group Γ :=π 1 (X)/H acts by deck transformations onX := X/H and that X =X/Γ. Thus, summarising: (X, dX) is a topological manifold with the integral homology of a point and the action of Γ onX has no fixed points.
In order to prove that (X, d) is bi-Hölder homeomorphic to a flat torus and that m is a constant multiple of H N , it is enough to prove that Γ Z N . Since Γ is a finitely generated abelian group (recall Proposition 2.25), it is sufficient to show that Γ has no subgroup isomorphic to Z/pZ with p prime. This follows from (35): indeed, from Smith theory (see for instance [Bre72,Chap. 3]), if Z/pZ, with p prime, acts on a topological manifold with the mod p homology of a point then the set of fixed points is non empty.

APPENDIX: SOME BASIC PROPERTIES OF MGH APPROXIMATIONS
For the reader's convenience, in this appendix we recall some well known properties of mGH approximations used in the paper. We next estimate d X (z ′ , z). For this, by the definition of z ′ it is enough to estimate d X (z, x ′ ). We have: Thus, d X (z ′ , z) ≤ 3ε and d Y (ϕ(z ′ ), w) ≤ 7ε.
Step 3. Control of the measure distortion. Using that φ is an ε-GH approximation and the definition (37) of ϕ, it is clear that From the Bishop-Gromov volume comparison, we have that there existsC =C(K, N, V) > 0 such that