Semi-local simple connectedness of non-collapsing Ricci limit spaces

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $\epsilon>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+\epsilon)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.


Introduction
Studying Gromov-Hausdorff limits of manifolds with uniform curvature lower bounds has been very active and has applications in many different directions. With sectional curvature lower bounds, the limit spaces are known as Alexandrov spaces, whose geometrical and topological structures have been extensively studied. In particular, these limit spaces are locally contractible [Per1]. For Ricci curvature lower bounds, Cheeger, Colding, and Naber have developed a rich theory on the regularity and geometric structure of the Ricci limit spaces. On the other hand, surprisingly little is known about the topology of these spaces. In fact, it could be so complicated that even a non-collapsing Ricci limit space may have locally infinite topological type [Men]. About twenty years ago, Sormani and the second author [SW1,SW2] gave the first topological restriction, showing that the universal cover of any Ricci limit space does exist, but were not able to show that the universal cover is simply connected. Recall that a connected and locally path-connected topological space has a simply connected universal cover if and only if space itself is semi-locally simply connected. See [Sp,page 84] for an example whose universal cover exists but is not simply connected.
As the main result of this paper, we show that any non-collapsing Ricci limit space is semi-locally simply connected.
Theorem A. Let (M i , p i ) be a sequence of complete Riemannian n-manifolds converging to (X, p) in the Gromov-Hausdorff topology with Then X is semi-locally simply connected.
Theorem A implies that any non-collapsed Ricci limit space has a simply connected universal cover. Our proof does not depend on the results in [SW1,SW2].
In fact, we show that X is essentially locally simply connected and have a local version. To state the result precisely, we use the notion of module of 1-contractibility (see [Bor, Pet]).
G.W. is partially supported by NSF DMS 1811558. J.P. is partially supported by AMS Simons travel grant. 1 Definition 0.1. Let X be a metric space. For x ∈ X and t ≥ 0, we define ρ(t, x), module of 1-contractibility at x, as below: where B r (x) is the open metric ball of radius r centered at x.
From the definition, it is clear that X is semi-locally simply connected if for any x ∈ X, there is T > 0 such that ρ(T, x) < ∞; X is locally simply connected if for any x ∈ X, there is t i → 0 such that ρ(t i , x) = t i .
We state the local version of Theorem A with an estimate on ρ(t, x).
We explain some of the difficulties in studying semi-local simple connectedness. For Alexandrov spaces, Perelman showed there is a homeomorphism from the tangent cone at a point to a local neighborhood around this point [Per1] (also see [Ka]); together with the fact that tangent cones are metric cones [BGP], this leads to the local contractibility. However, for Ricci limit spaces, we no longer have such a connection between tangent cones and local topology. For non-collapsing Ricci limit spaces, Cheeger-Colding proved the important result that tangent cones at any point are all metric cones [CC1], but a neighborhood of a point could have infinite second Betti number [Men]. Also, tangent cones at a point may not be unique and may not be homeomorphic [CC2,CN]. Even at the fundamental group level, it is not clear how to connect the tangent cone to the neighborhood. From the point of view of [SW1], since the universal cover always exists, it remains to rule out a non-contractible loop that can be homotopic to loops lying in arbitrarily small metric balls. Such a loop cannot be lifted to an open path in the universal cover, so one can not use deck transformations to study them. Also, such a loop may have infinite length. The last option is to use the sequence. In order to pass local simply connectedness information from the sequence to the limit space, one needs uniform control on the module of 1-contractibility for the sequence (see Theorem 1.7). However, a sequence of manifolds with the conditions in Theorem A may not have uniform modules of 1-contractibility. In fact, examples of Otsu show that the sequence may have shorter and shorter nontrivial loops [Ot,page 262 Remark (2)].
The most important step in our proof of Theorem 0.2 is lim t→0 ρ(t, x) = 0. After proving that lim t→0 ρ(t, x) = 0 holds for all x, we can further improve the result to lim t→0 ρ(t, x)/t = 1, by using the structure of tangent cones, a modification of Sormani's uniform cut technique [Sor], and certain connection between the local fundamental group of the limit space and that of the sequence (see Section 5).
We outline our approach to prove lim t→0 ρ(t, x) = 0 as follows. For a point x in the limit space and a sequence of points x i on M i , we classify the limit points into three types based on the module of contractibility at, or around, x i . Roughly speaking, type I points are those that modules of 1-contractibility are uniformly controlled in a uniform neighborhood around all x i ; type II points are those that modules of 1-contractibility are not uniformly controlled at x i ; type III are the rest (see Definition 3.1 and Lemma 1.2). When x is a type I point, we can control ρ(t, x) by constructing a sequence of uniformly convergent homotopies (see Lemma 1.2 and Theorem 1.7); this is related to [Bor].
The proof of lim t→0 ρ(t, x) = 0 is an induction argument on the local volume around x. As the base case, we start with points whose local volume is strictly larger than the half volume of the same size ball in the corresponding space form, or for simplicity, points with half volume lower bound. We show that ρ(t, x i ) can be controlled by a linear function in this case, where x i ∈ M i converging to x; with this, we can deduce that lim t→0 ρ(t, x) = 0 holds (see Theorem 2.1 and Proposition 2.6). For the next induction step, we consider x with quarter volume lower bound. If x is of type I, then we are done. If x is of type II, we can use small loops around x i to construct a sequence of covering spaces of certain local balls B ǫ (x i ). On this sequence of covers, we can lift small loops as open paths. Moreover, these covers shall have half volume lower bound so that the base case in the induction can be applied. Anderson's results on small loops [An1] and Cheeger-Colding's volume convergence for non-collapsing Ricci limit spaces [Co,CC2] are essential in the above steps. The remaining type III case is the most technical situation. The method of type I points fails due to the lack of local control on 1-contractibility. Also, there are no small loops to construct local covers. The key observation is that, based on results on type II and the definition of type III, for any point x ∈ X, at least one of {ρ(t, x i )} i and ρ(t, x) is controlled, where x i ∈ M i converging to x ∈ X. This inspires us to construct each small piece of the desired homotopy from the data of the sequence M i or from that of the limit space X, in a delicate way (see more explanations in Section 4). With the result on type III points (Theorem 3.4), we can continue the induction argument and eventually finish the proof of lim t→0 ρ(t, x) = 0.
As a by-product from the study of points with half volume lower bound, we have the following. Corollary 0.3. Given n and L ∈ (1/2, 1], there is a constant C(n, L) such that the following holds.
Let M be a complete non-compact n-manifold of Ric ≥ 0. If M has Euclidean volume growth of constant L > 1/2, that is, for some p ∈ M , then for any x ∈ M , any r > 0, and any loop c in B r (x), c must be contractible in B Cr (x).
Li and Anderson independently showed that if M has Euclidean volume growth of constant ≥ L, then π 1 (M ) has order ≤ 1/L [Li,An2]; consequently, if L > 1/2, then M is simply connected. Hence Corollary 0.3 can be viewed as a quantitative description of the simple connectedness when L > 1/2. We mention that Corollary 0.3 holds for non-collapsing Ricci limit spaces as well (see Theorem 6.5). The simple connectedness of open non-collapsing Ricci limit spaces is previously known only when L is very close to 1 [Mu].
With Theorem A, we can naturally generalize structure result of fundamental groups of manifolds with Ricci curvature and volume bounded below to that of non-collapsing Ricci limit spaces (see Section 6).

Convergence of spaces with controlled 1-contractibility
We study Gromov-Hausdorff convergence of spaces with uniformly controlled module of 1-contractibility. This is related to [Bor], where Hausdorff convergence and the contractibility of subsets are considered.
By Definition 0.1, ρ(t, x) = L < ∞ means that for any σ > 0 and any loop c contained in B t (x), there is a homotopy between c and a trivial loop with the image of H contained in B L+σ (x). Throughout the text, we always use a term involving σ for this situation (for example, a term like σ2 −i ).
In general, ρ(t, x) may not be a continuous function. On the other hand, we can always bound ρ(t, x) by a so called indicatrix, which is continuous [Bor].
We always assume that T ∈ (0, 1) in this paper unless otherwise noted.
Conversely, suppose that (2) holds. Consider g(t) = sup α∈A ρ α (t), which satisfies lim t→0 g(t) = g(0) = 0 by assumption. By [Bor,Section 7], there is a concave and non-decreasing function λ(t) such that g(t) ≤ λ(t) on [0, T ). It is clear that λ(t) is continuous on (0, T ) because it is concave. Also, according to (35) in [Bor], lim t→0 λ(t) = 0 holds. Definition 1.3. Let ǫ > 0. We say that two loops c, c ′ : The lemma below illustrates a relation between Gromov-Hausdorff closeness and homotopies. The method is similar to [Tu, Pet]. Because we need this construction and its related estimates later, we include the proof for readers' convenience.
Lemma 1.4. Given T ∈ (0, 1), there is ǫ 0 = T /20 such that the following holds: Let (X, x) and (Y, y) be two length metric spaces with the conditions below: then there is a continuous map H ′ : S 1 × [0, 1] → X such that H ′ is a free homotopy in B 2 (x) between c ′ 1 and c ′ 2 . Proof. (i) Let c : [0, 1] → B 1 (y) be a loop. Because c is uniform continuous, we can choose a large integer N such that Next we connect q i to q i+1 by a minimal geodesic for each i and close it up as a loop by connecting q N −1 to q 0 by a minimal geodesic. In this way, we result in a loop c ′ , as a broken geodesic, in X. Note that for all i = 0, 1, ..., L − 1. Let l i be a minimal geodesic from c ′ (i/L) to c ′′ (i/L). Since the small loop (ii) We first define H ′ on the boundary S 1 × {0, 1} so that This implies for all t ∈ [0, 1] and s = 0 or 1, We choose a finite triangular decomposition Σ of S 1 ×[0, 1] such that diam(H(∆)) ≤ ǫ for any triangle ∆ of Σ. Let K 0 be the set of all vertices of Σ and let K 1 be the 1-skeleton of Σ. If v ∈ K 0 is in the boundary S 1 × {0, 1}, we have already defined If two vertices v and w of K 0 is connected by an edge that is not part of the boundary, then we connect H ′ (v) and H ′ (w) by a minimal geodesic in X. From this, we obtain a continuous map, which we still call H ′ , H ′ : K 1 → X. Let ∆ be a triangle of Σ and let ∂∆ be its boundary. By our construction, it is direct to check that if a triangle Σ does not have any boundary point as its vertex, then 1 where ∂∆ is the 1-skeleton of ∆; if Σ has one or more boundary points as vertices, then diam(H ′ (∂∆)) ≤ 15ǫ.
In particular, H ′ (∂∆) is contained in B 18ǫ (H ′ (v)) with 15ǫ < T , where v is a vertex of ∆. By assumption H ′ (∂∆), as a loop, is contractible in B 1 (v ′ ), thus we can extend the domain of the map H ′ from ∂∆ to ∆. Since the extension can be achieved over each ∆, we result in the desired free homotopy.
Remark 1.5. In Lemma 1.4(i), the existence of c ′ does not require condition (3).
Proof. We continue to use the notations in the proof of Lemma 1.4(ii). Note that by assumption (3'), for any σ > 0, there is a homotopy from H ′ (∂∆) to a trivial loop so that the image of the homotopy is contained in B λ(18ǫ)+σ (H ′ (v)). Thus for all z ∈ ∆, With Lemma 1.6, we show that locally controlled module of 1-contractibility is preserved under Gromov-Hausdorff convergence.
We state the conclusion of above theorem for B 3/2 (y) instead of B 1 (y), because we need some extra room for later use in Section 3.
The proof below is inspired by [Bor], where Hausdorff convergence and the contractibility of subsets, instead of loops, are considered. Besides these differences, compared with [Bor,Section 15], our statement is localized; also the proof is much simplified and streamlined with the length metric space condition. Later in Section 4, we will present a different proof of Theorem 1.7, whose strategy is more in line with our main construction.
Let q ∈ B 3/2 (y) and let c be a loop in B t (q). We will construct a homotopy between c and a trivial loop. For each j ≥ I, pick In particular, c j converges uniformly to c. Since the image of c I satisfies there is a homotopy H I : D → B λ(t+6ǫI )+σ·2 −I (q I ) between c I and a trivial loop, where D is the closed unit disk. By Lemma 1.6, we can construct a homotopy between c I+1 and a trivial loop In general, for j ≥ I + 1, suppose that we have constructed a homotopy H I,j between c j to a trivial loop with We apply Lemma 1.6 again to obtain This process gives us a sequence of homotopies {H I,j } ∞ j=I . Since for all z ∈ D and all j ≥ I, we conclude that as j → ∞, H I,j converges uniformly to a continuous map H I,∞ : D → Y , which is a homotopy between c and a trivial loop. Moreover, we have Noting that I can be arbitrarily large and loop c is arbitrary in B t (q), we see Remark 1.8. From the proof of Lemma 1.4 and Theorem 1.7, we see two methods to move homotopies. The first method works when the target space has local contractibility (Lemma 1.4): we dissemble the homotopy into small pieces, then map the 1-skeleton to the target space then obtain the homotopy via extensions. The second method works when there is a sequence of spaces with uniformly controlled local contractibility converging to the target space (Theorem 1.7): we transfer the homotopy along the sequence and pass it to the target space by uniform convergence.

Half volume lower bound and 1-contractibility
In this section, we show that if the local volume is strictly larger than the half volume of a same size ball in the corresponding space form, then the module of 1contractibility is controlled by a linear function. As one of applications, this control implies that lim t→0 ρ(t, x) = 0 holds for x in the limit space if x has a local half volume lower bound (see Proposition 2.6).
For κ ∈ R and r > 0, we denote the r-ball in the n-dimensional space form of curvature κ as B n r (κ).
Because of Otsu's example [Ot], if the half volume lower bound is replaced by a positive lower bound, then the conclusion of Theorem 2.1 would fail.
(1) Let (M, p) be a Riemannian n-manifold with the conditions in Theorem 2.1. Then every loop in B ǫ (p) is contractible in B R (p).
(1) To find ǫ(n, κ, ω, R), we argue by contradiction. Then we would have a sequence (M i , p i ) of for all R ∈ (0, 1). Passing to a subsequence, we assume (M i , p i ) GH −→ (X, p). Without lose of generality, we also assume that each γ i has length less than 4ǫ i .
Let (U i , y i ) be the universal covering space of (B R (p i ), p i ) with covering group H i = π 1 (B R (p i ), p i ). Let Γ i be the subgroup generated by γ i . By [An1], each Γ i is a finite group with order ≤ N (n, κ, ω, R) (The statement in [An1] is about compact manifolds; its proof extends clearly to local balls). Since d(γ i y i , y i ) ≤ 4ǫ i and γ i has order ≤ N for all i, it follows that where F i is the Dirichlet domain centered at y i . This clearly leads to a contradiction since 2ω > 1. We complete the proof of (1). ( . This means that we can find a sequence of n-manifolds (M i , p i ) with the conditions below: (ii') for each i there is a loop γ i contained in the ball of radius 2ǫ i R −1 i (→ 0) and γ i is not contractible in the concentric unit ball. We apply the argument used in (1) once again and result in a contradiction.
Proof of Theorem 2.1. Let Corollary 0.3 follows directly from Theorem 2.1.
Proof of Corollary 0.3. By relative volume comparison, holds for all x ∈ M and all R > 0. Together with Theorem 2.1, the result follows immediately.
where H n is the n-dimensional Hausdorff measure in X.
By relative volume comparison, this limit always exists and belongs to (0, 1]. In fact, it is clear that ω(x) has a uniform positive lower bound for all x ∈ B 1 (p).
Proof. For a fixed small ǫ > 0 with By volume convergence [Co,CC2], there is N > 0 such that for all i ≥ N , we have Put t = s 0 , thus for all i ≥ N and y ∈ B d0 (x i ), vol(B s0 (y)) ≥ ω ′ · vol(B n s0 (−1)) By Theorem 2.1, there are T, C > 0, depending on n, s 0 , and ω ′ , such that for any i ≥ N and any y ∈ B d0 (x i ), ρ(t, y) ≤ Ct holds for t ∈ [0, T ). By Theorem 1.7, in the limit we have ρ(t, x) ≤ Ct for t ∈ [0, T ).
Remark 2.7. By a similar volume estimate presented above, it is clear that if ω(x) > ω 0 , then there is r > 0 such that ω(y) > ω 0 for all y ∈ B r (x).
3. Classification of points by 1-contractibility We classify points in the limit space by the module of 1-contractibility from the sequence.
Definition 3.1. Let (X i , p i ) be a sequence of length metric spaces converging to (X, p). Let x ∈ X.
We say that x is of type I, if there is r > 0 such that the family of functions We say that x is of type III, if it is not of type I nor type II.
The following lemma assures that Definition 3.1 is well-defined because it does not depend on the choice of x i .
Proof. Suppose that the family {ρ(t, x i )} i is not equally continuous at t = 0, then there are ǫ > 0, t j → 0, and a subsequence x i(j) such that is not equally continuous at t = 0 as well. This completes the proof.
Note that by Definition 3.1, the set of type I points is open in X. Due to Otsu's example, type II points in general may exist under the assumption of Theorem A. For a type III point x and a sequence x i ∈ X i converging to x, by definition the family {ρ(t, x i )|i ∈ N} is equally continuous at t = 0. Since for any r > 0, the family {ρ(q, t)|q ∈ B r (x i ), i ∈ N} is not equally continuous at t = 0, this implies that the closure of B r (x) must contain a point of type II. Hence any type III point must be a limit point of the set of type II points.
Corollary 3.3. Let (X i , p i ) be a sequence of length metric spaces with the conditions below: (1) the closure of B 2 (p i ) is compact; Proof. Since x is of type I, by Lemma 1.2 there is an indicatrix λ(t) on [0, T ) such that for all i and all y ∈ B r (x i ), ρ(t, y) ≤ λ(t) holds on [0, T ). By Theorem 1.7, x satisfies ρ(t, x) ≤ λ(t) on [0, T ) and the result follows.
Also, for X with the conditions in Theorem 0.2, by the proof of Proposition 2.6, any point x ∈ B 3/2 (p) with ω(x) > 1/2 is of type I.
We will prove the theorem below on type III points in Section 4.
Theorem 3.4. Let (X i , p i ) be a sequence of length metric spaces with the conditions below: (1) the closure of B 2 (p i ) is compact; ( Assuming that Theorem 3.4 is true, then we can prove lim t→0 ρ(t, x) = 0 holds for all x ∈ B 1 (p) in the context of Theorem 0.2 (see Theorem 3.5 below). Later in Section 5, we will strengthen the conclusion to lim t→0 ρ(t, x)/t = 1, which completes the proof of Theorem 0.2.
Note that if Theorem 3.4 is true, then the statement also holds if one replace B 3/2 (p) and B 1 (p), by balls B r (z) and B 2r/3 (z), respectively, where r ∈ (0, 1) and B r (z) ⊂ B 3/2 (p). This follows directly from a rescale of the metric.
We choose a small r > 0 so that ω(z) > 2 −(k+1) for all z ∈ B r (x) ⊂ B 3/2 (p) (Remark 2.7). Suppose that lim t→0 ρ(t, x) > 0, by Corollary 3.3 and Theorem 3.4, then there must be a type II point z ∈ B r (x) so that lim t→0 ρ(t, z) > 0. Let z i ∈ M i converging to z. According to Definition 3.1, passing to a subsequence if necessary, there is t i → 0 and small ǫ > 0 such that ρ(t i , z i ) ≥ ǫ for all i. In other words, for each i there is a loop γ i contained in B ti (z i ) that is not contractible in B ǫ (z i ). Without lose of generality, we can assume that γ i is based at z i and has length ≤ 4t i . As we did in the proof of Lemma 2.3(1), we consider (U i , y i ) as the universal covering space of (B ǫ (z i ), z i ) with covering group H i = π 1 (B ǫ (z i ), z i ). Let Γ i := γ i . The sequence (U i , y i ) may not be precompact in Gromov-Hausdorff topology (see [SW2,Example 3.2]), so we will take a sequence of small balls around y i , which always has precompactness. We consider the pseudo-group action of H i on B ǫ/2 (y i ) ⊂ U i . Passing to a subsequence if necessary, we obtain the equivariant Gromov-Hausdorff convergence below: where H acts isometrically and B ǫ/2 (z) is isomorphic to B ǫ/2 (y)/H [FY]. As seen in Lemma 2.3, #Γ i ≤ N for some N and diam(Γ i · y i ) → 0. Consequently, G is a finite group whose action fixes y. Let F i be the fundamental domain centered at y i . For any s ∈ (0, ǫ/4), we estimate: By volume convergence [Co,CC2], Thus By the induction assumption, we deduce that lim t→0 ρ(t, y) = 0. For any loop c in B t (z) based at z = π(y), connecting z to a point in c by a minimal geodesic if necessary, we can assume that c is based at z. We lift c to a curve c starting at y [Bre, Chapter II Theorem 6.2]. Note that the orbit H · y is discrete. Let d > 0 the distance between y and its closet point in the orbit H · y. For t < d/4, the liftc is contained in B t (H · y); hence it must be contained in B t (y) becausec is a curve. Since y is the only orbit point in B t (y), the lift c ends at y as well. Moreover, because c is contained in B t (y), for any σ > 0 there is a homotopy between c and a trivial loop with image in B ρ(t,y)+σ (y). Projecting this homotopy down to B ǫ (z), we obtain a desired homotopy from c to a trivial loop with image in B ρ(t,y)+σ (z). Since σ and loop c in B t (z) is arbitrary, it follows that lim sup t→0 ρ(t, z) ≤ lim sup t→0 ρ(t, y) = 0.
This completes the proof.
For the rest of this section, we prove some lemmas on the type III points, which will be used in the next section. We always assume the hypothesis of Theorem 3.4 without mentioning, that is, we assume that lim t→0 ρ(t, x) = 0 holds for all type II points.
For any L > 0, we define where B 1.2 (p) means the closure of B 1.2 (p). It follows from the hypothesis and Corollary 3.3 that Ω(L) is a subset of the set of type III points in B 1.2 (p). If Ω(L) is empty for all L > 0, then the conclusion of Theorem 3.4 clearly holds. Hence we will assume that Ω(L) is non-empty for some L > 0. In particular, w is of type III.
Proof. Let {z j } j be a sequence in Ω(L) converging to a point w. Suppose that Let ǫ > 0 with l w + 2ǫ < L. Let δ ∈ (0, ǫ) such that This shows that for j large, lim sup t→0 ρ(t, z j ) ≤ l w + 2ǫ < L.
A contradiction. By Corollary 3.3 and the hypothesis on type II points, w must be type III.
Then the family of functions is equally continuous at t = 0. Consequently, there is an indicatrix Λ(t) on [0, T ) such that ρ(t, w) ≤ Λ(t) for all t ∈ [0, T ) and all ρ(t, w) in the family.
Proof. Suppose the contrary, then there is ǫ 0 > 0, t j → 0, and z i(j) ∈ B 3/2 (p i ) such that where each z i(j) is ǫ i(j) -close to a point q j in Ω(L). Passing to a subsequence if necessary, we assume that q j converges to q ∈ Ω(L), the closure of Ω(L). Since z i(j) → q as j → ∞, it follows that q is of type II, which is a contradiction to Lemma 3.6.
Thus for any fixed y, there is T y > 0 so that ρ(T y , y) < L + σ.
To find a uniform T > 0 that works for all y ∈ A, we argue by contradiction.
Suppose that there are sequences T j → 0 and y j ∈ A satisfying Passing to a subsequence if necessary, y j converges to a point w. Note that In other words, w ∈ Ω(L), which contradicts the fact that all y j are at least distance r away from Ω(L) for all j.

Constructing homotopies
We first roughly explain our approach to prove Theorem 3.4. Lemma 3.7 says that for points z i ∈ X i that is close to z ∈ Ω(L), we have uniform control on module of 1-contractibility of z i ; Lemma 3.8 says that for a limit point y away from Ω(L), we have control on ρ(t, y). For a loop c contained in a small ball of x, using the method in Lemma 1.4, we can construct a continuous map H : K 1 1 → X defined on a 1-skeleton of D extending c. Ideally, for a triangle H(∂∆) outside Ω(L), we wish to extend H over this triangle right away by Lemma 3.8; for a triangle inside Ω(L), we wish to construct the homotopy from the sequence by utilizing Lemma 3.7. In other words, part of the desired homotopy comes from the limit space, while the other part comes from the sequence. When combining these two procedures together to construct the homotopy, we also need to be cautious to assure that we end in a continuous map with controlled size. In practice, we will indeed consider a sequence Ω(L j ) instead of a single Ω(L), where L j → 0.
Regarding constructing the homotopy from the sequence, recall that the proof of Theorem 1.7 has a similar fashion (also compare Lemma 3.7 with the conditions of Theorem 1.7). We take a close look at whether the proof of Theorem 1.7 in Section 1 could be useful here. The strategy illustrated above involves a step determining which procedure to be applied for a triangle H(∂∆) based on its position. Also, eventually the desired homotopy will be an extension of H : K 1 1 → X according to this strategy. On the other hand, recall that in the proof of Theorem 1.7 in Section 1, we transferred a homotopy along the sequence {X j } j≥I and forced a uniform convergence, thus the proof does not involve any image of 1-skeleton in the limit space.
Because the method in proving Theorem 1.7 in Section 1 is not compatible with our strategy here, this motivates us to write a new proof of Theorem 1.7, which has the advantage that we can keep the image of 1-skeletons in the limit space at each step. In this alternative proof, we will construct the desired homotopy by defining it on finer and finer skeletons of D. As indicated, this method also constitutes part of our proof of Theorem 3.4.
We now establish a lemma on constructing a homotopy through refining skeletons. This is similar to [Per2,5.1].
Lemma 4.1. Let D be the closed unit disk and let (X, p) be a metric space with the closure of B 2 (p) being compact. Let Σ j be a sequence of finite triangular decompositions of D with the conditions below: (1) each Σ j+1 is a refinement of Σ j , (2) diam(∆) ≤ j −1 for every triangle ∆ of Σ j . Suppose that we have a sequence of continuous maps G j : for any z ∈ K 1 j+1 − K 1 j , d(G j+1 (z), G j (u)) ≤ 2 −j holds for all u in the boundary of ∆, where ∆ is a triangle of Σ j containing z. Then {G j } j converges to a continuous map G ∞ : D → B 2 (p).

Proof. Clearly we can define
j . It suffices to show that G ∞ is uniform continuous. If true, then we can extend G ∞ continuously over D.
First notice that condition (4) Let ǫ > 0. We choose a large integer J so that 2 −J+1 ≤ ǫ/3. From the triangular decomposition Σ J , we construct an open cover U of D as follows. For every triangle ∆ of Σ J , we choose a connected open neighborhood U ∆ of ∆ such that all vertices in U ∆ belong to ∂∆. In this way, if y ∈ U ∆ , then there is u ∈ ∂∆ so that u and y lies in a common triangle of Σ J (this common triangle may not be ∆). Let U be the collection of all these U ∆ , and let δ > 0 be a Lebesgue number of the open cover U. For any y 1 , y 2 ∈ ∪ ∞ k=1 K 1 k , if d(y 1 , y 2 ) < δ, then there is a triangle ∆ of Σ J so that y 1 , y 2 ∈ U ∆ . Let u i ∈ ∂∆ such that u i and y i lies in a common triangle of Σ J (i = 1, 2). Then This completes the proof.
We can use the construction in proof of Lemma 1.4(ii) to define an extension on the 1-skeleton, with the distance estimate below.
Lemma 4.2. Let (X, x) and (Y, y) be two length metric spaces. Let c(t) and c ′ (t) be two loops in X and Y , respectively. Suppose that (1) the closure of B 2 (p) is compact, where p = x or y; (2) d GH ((X, x), (Y, y)) ≤ ǫ, (3) c and c ′ are 5ǫ-close, Then there is are triangular decomposition Σ of D and a continuous map If condition (5) is replaced by (5') c is contractible in B ρ (c(0)) via a homotopy H, then we have (E2) d(H ′ (z), H ′ (u)) ≤ ρ + diam(c) + 32ǫ, for all z ∈ K 1 − ∂D and u ∈ ∂D.
Proof. We choose a triangular decomposition Σ of D so that diam(H(∆)) ≤ ǫ for every simplex ∆ of Σ. We follow the method in the proof of Lemma 1.4(2) to construct H ′ : first define H ′ on ∂D by sending c(t) to c ′ (t), then define H ′ on 0-skeletons by mapping any vertex to a near by point, then on edges by minimal geodesics. Then from the proof of Lemma 1.4(2), we have diam(H ′ (∂∆)) ≤ 15ǫ, for all triangle ∆ of Σ. When z ∈ ∂D, then clearly d(H(z), H ′ (z)) ≤ 5ǫ.
When z ∈ K 1 −∂D, let ∆ be a triangle containing z and let v be a vertex connected to z, then Also, for any z ∈ K 1 − ∂D, we have (E1): If alternatively, c is contractible in B ρ (c(0)) instead of B ρ (x), then for any z ∈ K 1 − ∂D and u ∈ ∂D, we have (E2): We use Lemmas 4.1 and 4.2 to present an alternative proof of Theorem 1.7.
Alternative proof of Theorem 1.7. Let σ > 0. Let q ∈ B 3/2 (y) and c be a loop in B t (q), where t ∈ (0, T ). Let ǫ i → 0 be a decreasing sequence with ǫ 1 ≤ T /20, ǫ i+1 ≤ ǫ i /20, and λ(20ǫ i ) + 20ǫ i ≤ 2 −i for all i. We fix a large integer I so that Let c I be a loop in X I that is 5ǫ I -close to c. By hypothesis, c I is contractible in B λ(t+10ǫI )+σ2 −I (c I (0)) via a free homotopy H I . Following the procedure and estimates in Lemma 4.2, there is a triangular decomposition Σ 1 of D and a continuous map, which corresponds to H ′ as in Lemma 4.2, (G 1 (z), q) ≤ λ(t + 10ǫ I ) + σ2 −I + 18ǫ I for any z ∈ K 1 1 (Lemma 4.2(E1)). Next we apply the same procedure to every loop G 1 (∂∆), where ∆ is any triangle of Σ 1 . Let c ∆ be a loop in X I+1 that is 5ǫ I+1 -close to G 1 (∂∆). Then By assumption, c ∆ is contractible in B λ(16ǫI )+σ2 −I−1 (c ∆ (0)) with a homotopy H ∆ . The same procedure provides Σ 2,∆ , a triangular decomposition of ∆, and a continuous map G 2,∆ : for all z ∈ K 1 2,∆ and u ∈ ∂∆. Since this can be done for each triangle ∆ of Σ 1 , we obtain Σ 2 , a triangular decomposition of D which refines Σ 1 , and a continuous map G 2 : K 1 2 → Y such that d(G 2 (z), G 1 (u)) ≤ 2 −I (1 + σ/2) holds for all z ∈ K 1 2 − K 1 1 and all u ∈ ∂∆, where ∆ is a triangle of Σ 1 containing z. Repeating this process, we result in a sequence of triangular decomposition Σ j and a sequence of continuous maps G j : for all z ∈ K 1 j+1 − K 1 j and all u ∈ ∂∆, where ∆ is a triangle of Σ j containing z. By Lemma 4.1, G j converges to a continuous map G ∞ : D → Y , which realizes the homotopy between c and a trivial loop. Moreover, by (1B) and (jB), we have (1 + σ/2)2 −(I+j−1) < λ(t + 10ǫ I ) + 20ǫ I + (2 + 2σ)2 −I for all z ∈ D. In other words, im(G ∞ ) ⊂ B λ(t+10ǫI )+20ǫI +(2+2σ)2 −I (q).
Since I can be chosen arbitrarily large and loop c is also arbitrary in B t (x), we conclude that ρ(t, q) ≤ λ(t).
As indicated, when constructing the homotopy in proving Theorem 3.4, we will extend some of the pieces directly in the limit space by using Lemma 3.8. To accommodate this procedure, we modify Lemma 4.1 as below so that we can fill in some of the triangles at every step.
Lemma 4.3. Let D be the closed unit disk and let (X, p) be a metric space with the closure of B 2 (p) being compact. Let L j → 0 be a sequence of positive numbers. Let (Σ 1 , G 1 , E 1 , F 1 ) be a quadruple defined as below: (1A) Σ 1 , a finite triangular decomposition of D; (1B) G 1 : K 1 1 → B 1 (p), a continuous map defined on the 1-skeleton of Σ 1 ; (1C) E 1 , a collection of some triangles of Σ 1 ; Then there is a continuous map H : D → B 2 (p) that extends G j and F j for all j.
Proof. It is clear that at each step, {G k } j k=1 and {F k } j k=1 form a continuous map H j : ∪ j k=1 (K 1 k ∪ E k ) → B 1 (p). We show that this sequence {H j } naturally defines a uniform continuous map . If true, then H ∞ extends continuously over D.
Let ǫ > 0. Choose an integer J so that 2 −J ≤ ǫ/16 and L j ≤ ǫ/8 for all j ≥ J. From ∪ J k=1 Σ k , a triangular decomposition of D, we construct an open cover U of D as we did in the proof of Lemma 4.1. Let δ 1 > 0 be a Lebesgue number of U. Let δ 2 > 0 so that d(H J (y 1 ), H J (y 2 )) ≤ ǫ/4 for all y 1 , y 2 ∈ ∪ J k=1 (K 1 k ∪ E k ) with d(y 1 , y 2 ) ≤ δ 2 . We put δ = min{δ 1 , δ 2 }. Let ∆ J be any triangle of Σ J in C J−1 . We claim that holds for any y ∈ ∆ J ∩ (∪ ∞ k=J (K 1 k ∪ E k )) and any v ∈ ∂∆ J . In fact, there are two cases on how H ∞ (y) is defined: Case 1. H ∞ (y) is defined as G j (y) for some j ≥ J. Then by condition (jB), for any v ∈ ∂∆ J , if j > J, then by condition (jB), for any v ∈ ∂∆ J , we have

This verifies the claim.
Let y 1 , y 2 ∈ ∪ ∞ j=1 (K 1 j ∪ E j ) with d(y 1 , y 2 ) ≤ δ. If both y 1 and y 2 belong to ∪ J k=1 E k , then d(H ∞ (y 1 ), H ∞ (y 2 )) = d(H J (y 1 ), H J (y 2 )) ≤ ǫ/4 because δ ≤ δ 2 . If both y 1 and y 2 belong to C J = D − ∪ J k=1 E k , since δ ≤ δ 1 , there is a triangle ∆ J of Σ J with y 1 , y 2 ∈ U ∆J . Let v i ∈ ∂∆ J so that v i and y i lies in a common triangle of Σ J (i = 1, 2). Then by the claim we have shown, Finally, if y 1 ∈ ∪ J k=1 E k while y 2 ∈ C J , there is a point v on the segment from y 1 to y 2 so that v ∈ C J ∩ (∪ J k=1 E k ). Since d(y 1 , v) ≤ δ and d(y 2 , v) ≤ δ, we see that We complete the proof of uniform continuity.
Proof of Theorem 3.4. Let x be any type III point in B 1 (p). Let for all i and all t ∈ [0, T ). Shrinking T if necessary, we can assume that λ(T ) < 1/40. Suppose that lim t→0 ρ(t, x) = 0 fails for this x, then Ω(L) is non-empty for some L > 0; we will show that ρ(t, x) ≤ λ(t) for all [0, T ), which is a contradiction.
Let L j → 0 be a decreasing sequence. For each j, we define By Lemma 3.7, for each j there is an indicatrix Λ j (t) that bounds all ρ(t, z), where z ∈ B 1.2 (p i ) is ǫ i -close to some point w of Ω(L j ). Since lim t→0 Λ j (t) = 0 for each j, we can choose s j > 0 with Λ j (2s j ) + 4s j ≤ 2 −j .
We put A j := A(Ω(L j ); s j , 1.1). By Lemma 3.8, for each j, there is T j > 0 such that ρ(T j , y) ≤ L j + 2 −j for all y ∈ A j . Replacing T j by a smaller number if necessary, we can assume that for all j. We choose a subsequence i(j) so that Let σ > 0. Fix t ∈ (0, T ) and a loop c contained in B t (x). We will construct a homotopy between c and a trivial loop with controlled image by using Lemma 4.3. Fix an integer J ≥ 1 so that ). By the construction in Lemma 4.2 (also see the procedure in alternative proof of Theorem 1.7), there is a triangular decomposition Σ 1 of D and a continuous map 1 (Lemma 4.2(E1)). In particular, im(G 1 ) belongs to B 1/20 (x) ⊂ B 1.1 (p). Note that if a triangle ∆ has a point u ∈ ∂∆ so that G 1 (u) ∈ A J , then by (1A), G 1 (∂∆) is contractible in B LJ +2 −J (G 1 (u)). In this case we can directly extend G 1 continuously over ∆. With this in mind, we consider (1C) E 1 = {z ∈ D| z ∈ ∆ with G 1 (∂∆) ∩ A J = ∅}. As explained, over E 1 , G 1 extends to a continuous map F 1 : E 1 → X. (1D) F 1 : E 1 → X satisfies that for any ∆ ⊂ E 1 , there is u ∈ ∂∆, such that d(F 1 (z), G 1 (u)) ≤ L J + 2 −J ≤ L J + 2 −J holds for all z ∈ ∆. This completes the first step in constructing the desired homotopy.
We repeat this procedure for each k ≥ 2. This allows us to construct the quadruple (Σ k , G k , E k , F k ) with (kA) Σ k , a triangular decomposition of C k−1 that refines Σ k−1 | C k−1 , where holds for all z ∈ ∆ ∩ (K 1 k − K 1 k−1 ) and all u ∈ ∂∆; With (kB), we can check that im(G k ) does not exceed B 1.1 (p): Then we define the followings.
holds for all z ∈ ∆.
Applying Lemma 4.3, we end in a continuous map H extending all G k and F k . H realizes the homotopy between c and a trivial loop. Moreover, by the above (1B), (kB), and (kD), we see that for any z ∈ D, Since J can be arbitrarily large, it follows that ρ(t, x) ≤ λ(t). This completes the proof.
5. Proof of lim t→0 ρ(t, x)/t = 1 Based on Theorem 3.5, we finish the proof of Theorem 0.2 in this section by using Sormani's uniform cut technique [Sor].
The uniform cut technique is based on Abresh-Gromoll's excess estimate [AG]. Even though excess estimate has been extended to metric spaces with synthetic Ricci curvature bounds (RCD spaces) [GM, MN], which includes Ricci limit spaces, it is unclear to the authors whether excess estimate holds on a covering space of a local incomplete ball in the limit space. Therefore, we will go back to the sequence, find uniform cut points on the manifolds, then pass them to the limit.
We assume dimension n ≥ 3 in this section. When n = 2, the limit space is an Alexandrov space, which is locally contractible.
Our first goal is a localized version of uniform cut theorem with a parameter ǫ (see Lemma 5.3). This is similar to [Wy,Section 4], where the nonnegative Ricci curvature case is considered. We include the complete proof for readers' convenience.
Proof. Suppose that d(x, γ(d/2)) < ψ(ǫ)d, where ψ(ǫ) as given in the statement; we will determine the constant C 1 (n) in the end. Applying Theorem 5.1, On the other hand holds.
Lemma 5.3. Let (M n , p) be a manifold of n ≥ 3, Ric ≥ −(n − 1), and the closure B 2 (p) being compact. Let x ∈ B 1 (p). Let ǫ > 0 and let ψ(ǫ) be the constant in Lemma 5.2. Suppose that γ is a geodesic loop in π 1 (B R (x), x) of length d with the properties below, where B R (x) ⊂ B 1 (p).
(1) If a loop γ ′ based at x is homotopic to γ then γ ′ has length ≥ d.
Proof. Suppose that d(y, γ(d/2)) < ψ(ǫ)d. Let σ be a minimal geodesic from γ(d/2) to y. Let (U,x) be the universal cover of (B R (x), x), wherex is a lift of x in U . We lift γ to a minimal geodesicγ starting fromx in U . Since im(σ) ⊂ B R (p), we can also lift σ to a curveσ fromγ(d/2) to a pointỹ. It is clear that which is a contradiction sinceσ has length < ψ(ǫ)d.
Let x ∈ B 1 (p) as in Theorem 0.2 and let B R (x) ⊂ B 1 (p). From Theorem 3.5, B R (x) is semi-locally simply connected. As a result, the universal cover of B R (x) exists and π 1 (B R (x), x) acts freely and isometrically on the universal cover. We denote this universal cover as (U,x). If π 1 (B R (x), x) is not trivial, then we define the minimal length d = min and a subset S by Let h ∈ S and let γ be a minimal geodesic fromx to hx. It is not difficult to show that γ, the projection of γ to B R (x), is a geodesic loop and satisfies a halfway property [Sor], namely, γ is minimal on [0, d/2] and [d/2, d]. Ideally, we want to find a sequence of geodesic loops γ i converging to γ so that each γ i satisfies the halfway property as well. However, this may not be true because γ i may not be the geodesic loop of minimal length and may not be a short generator. Also, M i may contain shorter and shorter loops disappearing in the limit. To overcome this, roughly speaking, we will consider all the geodesic loops converging to some element of S, instead of a fixed element γ ∈ S (see Lemma 5.5).
For Lemmas 5.4 and 5.5 below, we assume that x ∈ B 1 (p) as in Theorem 0.2. For B r (x) ⊂ B R (x) ⊂ B 1 (p), where d/2 < r < R, we set ρ = (R − r)/4. Lemma 5.4 below can be viewed as a localized version of a proof in [Tu].
Lemma 5.4. For any i sufficiently large, there is a group homomorphism Moreover, Φ i is onto G(r), where G(r) is the subgroup of π 1 (B R (x), x) generated by loops based at x and contained in B r (x).
Proof. Since lim t→0 ρ(t, x) = 0 for all y ∈ B 1 (p), by the compactness of B r+2ρ (x), we can choose T > 0 such that ρ(T, y) ≤ (R − r)/5 < ρ for all y ∈ B r+2ρ (x); in other words, any loop in B T (y) is contractible in B r+3ρ (x) ⊂ B R (x). We set ǫ 0 = T /20 and N large so that Let c i be a loop based at x i with im(c i ) ⊂ B r+ρ (x i ). Following the method in Lemma 1.4(i), we can construct a loop c based at x, as a broken geodesic loop, such that c is 5ǫ i -close to c i and Also, by the proof of Lemma 1.4(ii), if c ′ i , another loop based at x i contained in B r+ρ (x i ), is homotopic to c i in B r+ρ (x i ), then the corresponding c ′ is homotopic to c in B R (x). It is clear that this defines a group homomorphism It remains to prove that Φ i is onto G(r). Let [c] be a generator of G(r). Because B R (x) is semi-locally simply connected, we can represent [c] by a geodesic loop c : [0, 1] → B r (x) based at x. Let 0 = t 0 , t 1 , ..., t k = 1 be a partition of [0, 1] so that c| [tj,tj+1] has length ≤ ǫ. Choose points z i,j in B r+ǫi (x) that is ǫ i close to c(t j ) (we set z i,0 = z i,k = x i ) and then connect these points by minimal geodesics. With this, we result in a broken geodesic loop c i in B r+ρ (x). It is direct to check Φ i ([c i ]) = [c] by our construction and Lemma 1.4(i).
Lemma 5.5. For each i, let S i = Φ −1 i (S), that is, the set of all the elements in π 1 (B r+ρ (x), x) mapped to S under Φ i . Let γ i be the unit speed geodesic loop of minimal length among all the geodesic loops representing elements of S i . Suppose that γ i has length d i . Then Proof. (1) Passing to a subsequence if necessary, geodesic loops γ i converge uniformly to a limit geodesic loop γ ∞ based at x. By construction of Φ i and Lemma 1.
∈ S for all i large. γ ∞ has length l(γ ∞ ) at most lim inf i→∞ d i . Let γ be a geodesic loop based at x with length d and [γ] = [γ ∞ ]. Since γ has the shortest length among all non-contractible loops in π 1 (B R (x), x), we deduce that It remains to show that d ≥ lim sup i→∞ d i . We prove this by contradiction. Suppose that there is δ > 0 such that Passing to a subsequence, we have d + δ < d i for i large. Further shrinking δ if necessary, we can assume that any element of π 1 (B R (x), x) outside S ∪ {e} has length at least d + δ (we can assume so because π 1 (B R (x), x) acts on the universal cover U discretely). For γ we chosen above, we can follow Lemma 1.4(1) to obtain a sequence of loops α i in X i that is 5ǫ i -close to γ, where In particular, α i converges uniformly to γ. Since γ is contained in B d/2 (x), we have α i contained in B d/2+6ǫi (x i ). Also, due to our construction, it is clear that α i has length l(α i ) ≤ 3d. For each α i , we divide α i into N i pieces α i | [ti,j ,ti,j+1] such that each piece has length between δ/4 and δ/2. Note that N i ≤ 12d/δ. Passing to a subsequence, we can assume that all N i are equal. For each j = 0, .., N − 1, let β i,j be a loop joining a minimal geodesic from x i to α i (t i,j ), α i | [ti,j ,ti,j+1] , and a minimal geodesic from α i back to x i . β i,j has length l(β i,j ) ≤ 2(d/2 + 6ǫ i ) + δ/2 = d + δ/2 + 12ǫ i .
As i → ∞, each β i,j converges to a loop β j based on x with β j has length ≤ d+ δ/2. By the choice of δ, each Φ i ([β i,j ]) = [β j ] either is trivial or belongs to S. If all [β j ] are trivial, then would be trivial too, a contradiction. Consequently, there must be some [β j ] ∈ S. For such a [β j ], we have [β i,j ] ∈ S i , but for i large, This is a contradiction to our choice of γ i . This completes the proof of (1).
(2) Let σ i be a minimal geodesic from x to γ i (d i /2). Suppose that σ i has length < d i /2. Let c i,1 be the loop joining γ i | [0,di/2] and σ −1 i and let c i,2 be the loop joining Since both c 1 and c 2 have length strictly shorter than γ i , due to our choice of γ i , it follows that Φ i [c 1 ] and Φ i [c 2 ] do not belong to S. Each c i,j consists of two geodesics, thus With the above preparations, now we are ready to prove Theorem 0.2, that is, lim t→0 ρ(t, x)/t = 1. Suppose that Theorem 0.2 fails, then we can use Lemmas 5.5 and 5.3 to find uniform cut points on manifolds, then pass these uniform cut points from the manifolds to a tangent cone of the limit space, which would end in a contradiction to the structure of tangent cones. This argument is a modification of [Sor].
Proof of Theorem 0.2. Fix x ∈ B 1 (p). Suppose that there are δ > 0 and r j → 0 such that lim j→∞ ρ(r j , x)/r j ≥ 1 + δ. Let R j = (1 + δ)r j . Then the following holds: (1) each B Rj (x) is not simply connected, (2) elements in S j , the set of shortest nontrivial loops in π 1 (B Rj (x), x), are represented by loops contained in B rj (x).
Choose ǫ > 0 sufficiently small so that ǫ < ψ(ǫ) < δ/4, where ψ(ǫ) is given in Lemma 5.2. With this ǫ, where d j is the length of elements in S j . For each fixed j, by Lemma 5.5, there is γ i,j of length d i,j with halfway property and d i,j → d j as i → ∞. Let m i,j = γ i,j (d i,j /2) be the midpoint of γ i,j . Since R j > (1/2+2ψ(ǫ))d i,j for all i large, applying Lemma 5.3, we have d(y, m i,j ) ≥ ψ(ǫ)d i,j for all y ∈ ∂B (1/2+ǫ)di,j (x).
Next we consider the convergence: where C x X is a metric cone with vertex v since X is a non-collapsing Ricci limit space [CC1]. By a standard diagonal argument, we have a convergent subsequence With respect to the above convergence, m i(j),j → m ∈ C x X with d(m, v) = 1/2. We claim that there are no rays starting from v and going through m, which contradicts the fact that C x X is a metric cone with vertex v. Let y be any point in ∂B 1 (v) and let y j in d −1 i(j),j B 1 (x i(j),j ) converges to y. Clearly d(y j , x) = 1 + η j on d −1 i(j),j B 1 (x i(j) ), where η j → 0. Let z j be the point where a minimal geodesic from m j to y j intersects d −1 i(j),j ∂B (1/2+ǫ)d i(j),j (x i(j) ). Then on d −1 i(j),j B 1 (x i(j) ), d(y j , m i(j),j ) = d(y j , z j ) + d(z j , m i(j),j ) ≥ (1 + η j ) − (1/2 + ǫ) + ψ(ǫ) = 1/2 + η j + (ψ(ǫ) − ǫ) Theorem 6.3. Given n, κ, D, v, there are only finitely many isomorphic classes of fundamental groups among spaces in M(n, κ, D, v).
Even though there are only finitely many isomorphic classes of fundamental groups among spaces in M(n, κ, D, v), the stability result is not true. Namely, there are spaces in M(n, κ, D, v) which are arbitrarily Gromov-Hausdorff close but have different fundamental groups (see [Ot] or Remark 6.2 above).
Theorem 6.4. Given n, D, v, there are positive constants ǫ(n, D, v) and C(n, D, v) such that for any X ∈ M(n, ǫ, D, v), π 1 (X) contains a normal abelian subgroup generated by at most n elements of index ≤ C.
Proof. By Corollary 6.1, it suffices to prove the statement for manifolds. The manifold result was proved in [KL,Theorem 1.7]. Here we present a different proof by using a result from [PR].
Suppose the contrary, then we have a sequence of Riemannian manifolds M n i with Ric Mi ≥ −i −1 , diam(M i ) ≤ D, vol(M i ) ≥ v, but any abelian subgroup of Γ i := π 1 (M i , p i ) has index ≥ 2 i . Passing to a subsequence, we obtain equivariant Gromov-Hausdorff convergence By the compactness of X and Cheeger-Colding's splitting theorem [CC1], Y splits isometrically as R k × K, where K is compact. Since M i is non-collapsing, both X and Y has Hausdorff dimension n, and G is a discrete group. By [FY], there is a sequence of subgroups H i of Γ i such that and Γ i /H i is isomorphic to G/G 0 = G for all i large. By [PR,Theorem 0.8] (also see Lemma 2.16 and Theorem 2.17), H i must be trivial for all i large. Thus Γ i is isomorphic to G for all i large. We claim that G is virtually abelian. If the claim is true, then clearly the desired contradiction follows. Let p : Isom(R k × K) → Isom(R k ) be the natural projection. By generalized Bieberbach theorem [FY], p(G) contains a subgroup Z r of finite index, where r ≤ k. Consider the exact sequence 1 −→ kerp ∩ p −1 (Z r ) −→ p −1 (Z r ) p −→ Z r −→ 1.
Because kerp is a discrete subgroup of Isom(K), which is compact, kerp is finite. By Lemma 4.4 of [FY], p −1 (Z r ) contains an abelian subgroup generated by r element and of finite index. The claim now follows from the fact that p −1 (Z r ) has finite index in G.
Theorem 6.5. Let X ∈ M(n, 0, v). Suppose that X has Euclidean volume growth of constant ≥ L. Then π 1 (X) is finite of order ≤ 1/L. If L > 1/2, then any loop in B r (x) is contractible in B Cr (x), where x ∈ X and r > 0.
Proof. The proof goes the same as the one of [An2] and our Theorem 2.1.
Again this was only known before assuming the universal cover is simply connected. We give a complete proof for readers' convenience.
Proof. Let X be the universal cover of X and letx ∈ X. Since relative volume comparison holds on X, we know that H n (B r (x)) vol(B n r (κ)) is a non-increasing function in r. Together with the fact that the space form of constant curvature κ > 0 has diameter π/ √ κ, we see that diamX ≤ diam X ≤ π/ √ κ.
Theorem 6.7. Given n, v > 0, there is a constant C(n, v) such that the following holds. Let X ∈ M(n, κ, v) be a Ricci limit space and let C x X be a tangent cone of X at x ∈ X. Then C x X is a metric cone C(Z) with π 1 (Z) having order ≤ C(n, v).
Proof. By [CC1], C x X is a metric cone C(Z) with diam(Z) ≤ π. H n−1 (Z) ≥ v ′ for some v ′ depending on (X, x). Also, Ric Z ≥ n − 2 in the sense of [LV, St]; in particular, relative volume comparison holds on Z. Thus the result follows if Z is semi-locally simply connected. For a point (1, z) ∈ C(Z), by Theorem A, lim t→0 ρ(t, (1, z)) = 0. Choose a T small so that ρ(T, (1, z)) < 1/2. For any loop c in B T (z) ⊂ B T ((1, z)), there is a homotopy H contracts c so that im(H) does not contain the vertex of C(Z). We define a retraction R : C(Z) − {vertex} → Z by sending (t, z) to z. Retracting im(H) to Z via this map R, we conclude that c is contractible in Z.