The Poincar\'e Inequality does not improve with blow-up

For each $\beta>1$ we construct a family $F_\beta$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_\beta$ admits a $(1,P)$-Poincar\'e inequality if and only if $P>\beta$.


Introduction
Background. The abstract Poincaré inequality was introduced in [HK98] in the study of quasiconformal homeomorphisms of metric measure spaces where points can be connected by good families of rectifiable curves. The investigation of PIspaces, i.e. metric measure spaces equipped with doubling measures and which admit a (1, P )-Poincaré inequality for some P ∈ [1, ∞), has been object of intensive research.
One trend of investigation has focused on the infinitesimal structure of such spaces. For example, Cheeger [Che99] formulated a generalization of the classical Rademacher Differentiation Theorem which holds for PI-spaces and showed that in such spaces the infinitesimal geometry of Lipschitz maps is rather constrained. Moreover, this result has allowed to extend notions of differential geometry, like tangent and cotangent bundles, to a large class of nonsmooth spaces which includes Carnot groups [Jer86], spaces with synthetic Ricci lower bounds [Raj12], some inverse limit systems of cube complexes [CK15], and boundaries of certain Fuchsian buildings [BP99]. There are also more complicated examples which involve gluing constructions [HH00,HK98] and passing to subsets [MTW13]. However, the infinitesimal geometry of all these examples is rather special, in the sense that a generic tangent/blow-up is biLipschitz equivalent to a product of Carnot groups with an Recent examples of spaces which admit (1, P )-Poincaré inequalities but not (1, P − ∆)-Poincaré inequalities have been constructed in [MTW13,DS13]. However, such examples are rectifiable, and so do not provide new infinitesimal geometries. Of particular interest are the examples of [DS13] which show that Cheeger's minimal upper gradient depends on the choice of the exponent P (i.e. if one has a (1, P )-Poincaré inequality but not a (1, P − ∆)-Poincaré, the P -minimal upper gradient and the (P − ∆)-minimal upper gradient can be different). One may check that this is not the case for our examples; this is unavoidable in the context of taking blow-ups: this will be discussed in a forthcoming paper where we show that blow-ups of differentiability spaces are differentiability spaces.
Overview. We observed that to produce new examples for the infinitesimal geometry of PI-spaces one might consider an inverse limit of square complexes where the gluing locus has 0 1-capacity [CK15,Example 11.13]. However, such examples would have analytic and Assouad-Nagata dimension 2, and would not give access to the full range of exponents P c . Moreover, the arguments in [CK15] would not carry over and one would have to resort to modulus estimates.
We thus decided to obtain X as an asymptotic cone of a metric graph G so that the stability under blow-up would be already built in the model. Note that one might also realize X as an inverse limit of a system of metric measure graphs, but it would not satisfy the same axioms as the inverse systems in [CK15]. Specifically, Axiom (2) in [CK15], i.e. the requirement that simplicial projections are open, would fail and the analysis in [CK15] would not carry over.
In Section 2 we construct the graph G and the corresponding measure µ G in function of some parameters. The choices for the weights on the measure will produce the different measures µ Pc . We then make a study of the shape of balls.
In Section 3 we construct good quasigeodesics that connect pairs of points in G. For convenience, we focus on the construction of good walks.
Section 4 contains the technical part of the paper. We establish modulus estimates to prove/disprove the Poincaré inequality in G for a given choice of P . In this section we also recall the definition of modulus and a "geometric" characterization of the Poincaré inequality in terms of random curves.
In Section 5 we define asymptotic cones and complete the proof of Theorem 1.1. In passing information from G to X we take advantage of a discretization procedure in [GL14].
A future line of investigation will involve the study of examples where in G we have only one set Θ of labels: these possess worse connectivity properties than the examples discussed here.
Notational conventions. We use the convention a ≈ b to say that a/b, b/a ∈ [C −1 , C] where C is a universal constant; when we want to highlight C we write a ≈ C b. We similarly use notations like a b and a C b. In the following C often denotes an unspecified universal constant (that can change from line to line) which can be explicitly estimated. We use the notation E[ϕ] to denote the expectation of the random variable ϕ. The notation B(A, r) denotes a ball of radius R centred on the set A, i.e. the set of points p at distance < r from the set A.

Construction of doubling graphs
2.1. Choice of parameters. We choose some parameters to construct the metric space X: (P1): An integer N ≥ 2; (P2): A sequence of integers {m k } k : m k ∈ {2, · · · , N } for each k; (P3): Two finite sets of symbols Symb 1 , Symb 2 with # Symb 1 ≥ 3 and # Symb 2 ≥ 2. The sets Symb 1 and Symb 2 share one symbol {∅} which we will call the end symbol; the set Symb 1 contains another symbol {♠} which we will call the gluing symbol.
The space X will be kind of self-similar in the sense that in order to analyze its geometry we will use only a sequence of scales. We thus introduce the scales σ k = k j=1 m j . To construct the metric space X we will take an asymptotic cone of an infinite graph G, see Section 5. For most of the paper we will work with G, which is obtained as follows. We let Λ (resp. Θ) denote the set of labels on Symb 1 (resp. Symb 2 ), i.e. the infinite strings λ = {λ(n)} (resp. θ = {θ(n)}) where λ(n) ∈ Symb 1 (resp. θ(n) ∈ Symb 2 ) and λ(n) (resp. θ(n)) is eventually the end symbol. We now regard R as a graph whose vertices are the elements of Z; using the scales σ k we associate to each m ∈ Z an order ord(m) by the formula: (2.1) ord(m) = 0 if m = 0 max{k : σ k divides |m|} otherwise.
Definition 2.2. Consider the disconnected graph R × Λ × Θ and a vertex v = (m, λ, θ). We say that v is a gluing point of order t if ord(m) = t > 1 and at least some symbol in {λ(j)} j<t is not the gluing symbol. We say that v is a socket point of order t if ord(m) = t and λ(j) is the gluing symbol for j < t. Note that a vertex with ord(m) = 1 is always a socket point.
We consider on G the length metric where each edge has length 1. Points in G are then equivalence classes [(t, λ, θ)] of points (t, λ, θ) ∈ R × Λ × Θ. The quotient map R × Λ × Θ → G will be denoted by Q. The Q-image of a gluing point (resp. a socket point) will be called a gluing point (resp. a socket point) of G. Note that the projection R × Λ → R induces a 1-Lipschitz map π : G → R.
To analyze the shape of balls in G the following definitions are useful. Definition 2.3. To the sequence of scales {σ k } we associate the discretized logarithm lg : [0, ∞) → N as follows: Note that each vertex v ∈ G has the form [(k, λ)] where k ∈ Z, and ord(k) will be called the order of v.

Construction of walks.
A walk on G is a finite string on vertices and edges W = {w 0 e 1 w 1 · · · e l w l } where w i−1 and w i are the endpoints of e i for 1 ≤ i ≤ l. In the following we will often suppress the edges from the notation, i.e. simply write W = {w 0 w 1 · · · w l }; we will also say that W is a walk from w 0 to w l and that l is the length of W , which we will denote by len W . The starting point str W of W is w 0 and the end point end W of W is w l . Two walks W 1 , W 2 with end W 1 = str W 2 can ba concatenated to obtain a walk W 1 * W 2 .
We say that a walk W from x to y is geodesic if len W = d(x, y). This notion can be also extended to the case in which x and / or y are not vertices of G. In this case a geodesic walk from x to y is a geodesic walk from a vertex w x to a vertex w y such that: note that (2.7) implies len W = d(w x , w y ). A walk W = {w 0 w 1 · · · w l } is monotone increasing (resp. decreasing) if for 0 ≤ i ≤ l − 1 one has π(w i+1 ) > π(w i ) (resp. π(w i+1 ) < π(w i )).
We have preferred to introduce walks because they are more convenient than parametrized paths to describe the construction of quasigeodesics and random curves that we present later in the paper. In working with walks, it is important to keep track of the labels of their vertices and edges. Specifically, except for countably many points of G, the fibre Q −1 (x) is a singleton; the points x for which #Q −1 (x) > 1 are either gluing points or socket points. Note that if x is neither a gluing point nor a socket point, the labels λ x ∈ Λ and θ x ∈ Θ are well-defined as x = π(x), λ, θ for unique λ = λ x and θ = θ x . In particular, if e is an edge, all points in e, except possibly one of the vertices, have the same labels λ e and θ e . On the other hand, if x is a gluing point of order k, x is a vertex of G of the form π(x), λ, θ where: θ is uniquely defined, and λ(l) is uniquely defined for l = k. If x is a socket point of order k, then it is a vertex of G of the form π(x), λ, θ where: λ(l) is the gluing symbol for l < k, λ(l) is uniquely defined for l > k, and θ(l) is uniquely defined for l = k. Therefore, if x is either a gluing point or a socket point, at most one entry of each label λ(l) and / or θ(l) is not uniquely defined; in this case we will sometimes make an arbitrary choice and still write λ x (l) or θ x (l). Note also that if x is a gluing point Q −1 (x) has cardinality # Symb 1 , and if x is a socket point Q −1 (x) has cardinality # Symb 1 ×# Symb 2 . Sometimes we will say that λ is the Λ-label of an edge or vertex and that θ is the Θ-label of an edge or vertex.
The following Lemmas 2.8, 2.9, and 2.14 will be used to build quasigeodesics in Section 3 and to prove the Poincaré inequality in Section 4.
Lemma 2.8. Let (p, k) ∈ G × N, and let (λ, θ) denote the labels of one of the edges e incident to p. Then there is a constant C depending only on (P1)-(P3) such that there are monotone walks W + and W − satisfying: (1) W ± is a walk from p to v ± , where either v ± is a gluing point if some {λ(j)} j<k is not the gluing symbol, or is a socket point of order k; (4) All edges in W ± have the same labels (λ, θ).
Proof. We just build W + . Because p is incident to an edge with label (λ, θ) we have p ∈ Q(R × {λ} × {θ}), and thus we can find a monotone increasing walk W 0 ⊂ Q(R × {λ} × {θ}) which starts at p, has length len W 0 ∈ [σ k , 2σ k ], and ends at a vertex w 0 with ord(w 0 ) = 0. There is a uniform constant C ≥ 1 such that the set R ∩ [π(w 0 ), π(w 0 ) + Cσ k ] contains an integer t with ord(t) = k. Let v + be the vertex of Q(R×{λ}×{θ}) which projects to t. Then, if all the symbols {λ(j)} j≤k−1 equal {♠}, v + is a socket point of order k; otherwise v + is a gluing point of order k. Let W 1 ⊂ Q(R × {λ} × {θ}) be a monotone increasing walk starting at w 0 and ending at v + . Then W + is obtained by concatenating W 0 and W 1 .
For i ≥ 1 the walk W i is obtained from W i+1 as follows. The (backward) inductive assumption is that the last edge of W i+1 has label (λ (i+1) , θ) and that the last vertex v i+1 of W i+1 is either a gluing or a socket point of order i + 1. Note is then defined as a monotone increasing walk starting at v i+1 and ending in v i . We then let w τi = v i .
We complete the construction by producing W 0 as follows; we let λ (0) be the label such that: We then let v + = v 0 be the vertex of Q([π(p), ∞)×{λ (0) }×{θ}) such that π(v 0 ) = t 0 . The walk W 0 is then a monotone increasing walk joining v 1 to v 0 . We now explain how each property in the statement of this Lemma holds: (1): because v + = v 0 is a socket point of order k as ord(t 0 ) = k and the label λ (0) has its first k − 1 entries equal to {♠}; (2): because we have lenW σ k , len W i σ i for i ≥ 1 and len W 0 σ 1 ; (3): because the walksW , W k−1 , W k−2 , . . . , W 0 lie in Q(R × Λ × {θ}); (4): because of howW was constructed; The next Lemma 2.14 is proven like Lemma 2.9; the proof is omitted as it looks like the specular image of the previous one.

Construction of measures.
We now turn to the construction of the measure µ on G. One possibility is to take the pushforward under the quotient map Q : R × Λ × Θ → G of the measure which coincides with Lebesgue measure on each R× {λ}. For extra flexibility we choose a finite set of weights Weight = {w s } s∈Symb 1 ∪ Symb 2 subject to the restrictions w s > 0 and w {∅} = 1. For each λ ∈ Λ and θ ∈ Θ we denote by w(λ), w(θ) the associated weights: where the products in (2.24-2.25) are actually finite. We also use the notation w((λ, θ)) for the product w(λ)w(θ).
Definition 2.26. We denote by µ the measure on G which is the pushforward of the measure on R × Λ × Θ which coincides with w((λ, θ))L 1 on each R × {(λ, θ)}. Note that different choices of the weights in Weight will produce mutually singular measures on the asymptotic cone X, compare [Sch15].
In particular, if the m k are all equal to some m and if R ≥ 1, we have: w(λ(n), θ(n)); moreover, if all the weights are equal to 1 one has: Proof. For each pair of labels (λ, θ) let T (k) λ,θ be the set of labels that can be obtained from (λ, θ) by making the first k entries of λ and/or θ arbitrary. We then compute as follows: w(λ(n), θ(n)), (2.32) which gives (2.28). Now (2.29) follows from (2.28) and Lemma 2.16 by observing that for any C 0 there is a C(C 0 ) such that: If we assume that all the m k are equal to m, then the discretized logarithm lg is just log m up to a bounded additive error, and hence (2.30), (2.31) follow.

Construction of good walks
In this section we prove the existence of good walks between points in G. These walks correspond to quasigeodesics which are used to build the families of curves used to prove Poincaré inequalities. Let x, y ∈ G; choose labels (λ x , θ x ), (θ y , λ y ) such that x = π(x), λ x , θ x , y = π(y), λ y , θ y and the cardinality of the set: In the following C will denote a universal constant that can change from line to line and that can be explicitly estimated.
Definition 3.2. Given x, y ∈ G with d(x, y) > 1 a good walk W = {w 0 e 1 w 1 · · · e L w L } from x to y is a walk having the following properties: Proof. Let w 0 , w 1 be vertices of G with ord(w 0 ) = ord(w 1 ), then: Take a geodesic walk W from x to y. Then there are w j0 , w j1 ∈ W such that w j0 is either a gluing or a socket point of order max N(x, y) and w j1 is either a gluing or a socket point of order k; letW be a subwalk of W joining w j0 and w j1 , and observe that: Theorem 3.6. If lg d(x, y) ≥ k max = max N(x, y) there is a good walk W from x to y which has the following additional properties: there is a distinguished gluing or socket point w s(k) such that each edge e preceding w s(k) satisfies λ e (k) = λ x (k), and each edge e following w s(k) satisfies either λ e (k) = λ y (k) or λ e (k) = {♠}. Moreover, in this case all edges e satisfy θ e (k) = and λ e (k) = λ x (k), and each edge e following w s(k) satisfies θ e (k) = θ y (k) and either λ e (k) = λ y (k) or λ e (k) = {♠}. Moreover, the map k → s(k) is monotone increasing and the subwalk W k from w s(k) to w s(k+1) satisfies: (GWA2): The walk W satisfies: Proof. Without loss of generality we can assume π(x) ≤ π(y). If N(x, y) = ∅ then x, y lie in some Q(R × {λ} × {θ}) and the construction of the walk is immediate. Let w 0 be the vertex of G satisfying π(w 0 ) ∈ [π(x), π(x) + 1), (λ w0 , θ w0 ) = (λ x , θ x ) (if the labels for w 0 or x are not unique, one can choose them so that equality holds. Note that for a non-unique label (λ p , θ p ) only one entry (λ p (m), θ p (m)) is not uniquely determined). Order the elements of N(x, y) increasingly: (3.9) k 0 < k 1 < · · · < k q .
. The goal is to construct a walk W k0 of length comparable to σ k0 which allows to change the k 0 -th entries of the labels. We build W k0 in two parts W (−) k0 and W (+) k0 . We now consider the first case θ x (k 0 ) = θ y (k 0 ) which implies λ x (k 0 ) = λ y (k 0 ); by Lemma 2.8 we can find a monotone increasing walk W (−) k0 from w 0 to a gluing or a socket point v k0 is the first part of the walk W k0 and we let w s(k0) = v (−) k0 . Letλ w0 be the label which agrees with λ w0 except at the k 0 -th entryλ w0 (k 0 ) = λ y (k 0 ). The second part of the walk W (+) k0 is a monotone walk of length len W (+) k0 ∈ [1, σ k0 ] which terminates at a vertex of order 0 and whose edges have the same label (λ w0 , θ w0 ). We now consider the second case θ x (k 0 ) = θ y (k 0 ) which is slightly more complicated. By Lemma 2.9 we can find a label-nonincreasing monotone walk W k0 has order k 0 and for l > k 0 one has (λ(v k0 is the first part of the walk W k0 and we let w s(k0) = v (−) k0 . By Lemma 2.14 we find a label-nondecreasing monotone walk W k0 of order zero satisfying: k0 ; l)) = (λ(y; l), θ(y; l)) and for l > k 0 The construction continues by induction on k j , i.e. suppose we have constructed the subwalks W k0 , · · · , W kj which form the first part of W . The first part W We then let w s(kj+1) = v (−) kj+1 . By Lemma 2.14 we complete W kj+1 by finding a label-nondecreasing monotone walk W kj+1 ; l)) = (λ(y; l), θ(y; l)) and for l > When we reach j = q we have constructed the first part W (1) of the walk W . Property (GW3) is satisfied because W (1) is monotone increasing and the part of (GW2) concerning w 0 is also satisfied; the additional condition (GWA1) is also satisfied on W (1) , and needs only to be checked there because of the way in which we construct the second part W (2) of the walk.
There are two cases to consider to complete the proof.
kq and y belong to Q(R×{λ y }×{θ y }). Therefore, W (2) is constructed by taking a geodesic walk in Q(R × {λ y } × {θ y }) from v (+) kq to y. We need only to prove (GW1) which is a consequence of (GWA2): As π is 1-Lipschitz and as W is monotone increasing, we have len W ≥ π(y) − π(x) which completes the proof of (GWA2).
Theorem 3.17. If lg d(x, y) < k max = max N(x, y) then there is a good walk W from x to y which has the following additional property: there is a distinguished gluing or socket point u kmax ∈ W of order k max such that each edge e preceding u kmax satisfies λ(e; k max ) = λ(x; k max ) and each edge following u kmax satisfies λ(e; k max ) = λ(y; k max ). Moreover, in this case all edges e of W satisfy θ(e; k max ) = θ(x; k max ). On the other hand, if θ(x, k max ) = θ(y, k max ) there is a distinguished socket point u kmax such that each edge preceding u kmax satisfies (λ(e; k max ), θ(e; k max )) = (λ(x; k max ), θ(e; k max )) and each edge e following u kmax satisfies (λ(e; k max ), θ(e; k max )) = (λ(y; k max ), θ(y; k max )). Moreover, W can be decomposed into consecutive walks W x and W y where W x is a good walk from x to u kmax satisfying the conclusion of Theorem 3.6, and W y is a good walk from u kmax to y satisfying the conclusion of Theorem 3.6.
Proof. The construction in the cases θ x (k max ) = θ y (k max ) and θ x (k max ) = θ y (k max ) is essentially the same, and we thus discuss only the latter case. The properties of the labels (λ(e; k max ), θ(e; k max )) follow from the construction and Theorem 3.6. Take a geodesic walk W from x to y. Note that there must be a socket point u ∈ W of order k max so that: moreover, let U denote the set of socket points of order k max and let u kmax be an Let k ∈ N(x, u kmax ); then if k > k max a geodesic walk W from x to u kmax would pass through either a gluing or a socket point of order k and by Lemma 3.3 we would have: can take any value, and hence k < k max ; we can then take a geodesic walk from x to u kmax which must pass through either a gluing or a socket point of order k, and we apply Lemma 3.3 to conclude that: Thus we can apply Theorem 3.6 to obtain a good walk W x from x to u kmax . Note that (3.19) implies that (λ(u kmax ; l), θ(u kmax ; l)) = (λ(x; l), θ(x; l)) for l > k max ; in particular, as k max = max N(x, y), if k ∈ N(u kmax , y) we have k < k max . Let W be a geodesic walk from u kmax to y; then it must pass through either a gluing or a socket point of order k and Lemma 3.3 implies: (3.21) d(y, u kmax ) = len W ≥ σ k ; therefore, we can apply Theorem 3.6 to obtain a good walk W y from u kmax to y. For later reference, we also note here that: The walk W is obtained by concatenating W x and W y so that it satisfies (GWA3). Property (GW1) follows observing that: and using (3.22) to conclude that: Property (GW2) holds because it holds for W x and W y . We discuss property (GW3) in some cases. We will denote by C 1 ≥ 2 the constant in (GW3) provided by Theorem 3.6. In the following we use the notations k (x) max = max N(x, u kmax ) and k (y) max = max N(u kmax , y).
(Case 1,1): W x and W y are both monotone. Then W is monotone and (GW3) holds.
(Case 1,2): W x is not monotone and W y is monotone. As in Theorem 3.6 we de- and C is a universal constant. Let w ∈ W y and j y (w) denote the position of w in W y and j(w) the position in W . If j y (w) < 2 len W x , then j(w) ≤ 3 len W x and so: (Case 1,3): Suppose that W x is monotone but W y is not. As in Theorem 3.6 we decompose W y in a first part W (m) y which is monotone and a second part we obtain (GW3) as in (Case 1,1). Note that: where in ( * ) we used (3.22) and where the constant in the lower bound can be explicitly estimated in terms of C 1 . Suppose that π(z i ) ∈ π(u kmax ), π(u kmax ) + σ k (y) max /2 . Then any geodesic walk from x to z i must pass through some socket pointũ ∈ U, and we would also have k The bounds (3.31), (3.32) imply that (GW3) holds on {z 0 , · · · , z m } with a constant that can be computed in terms of C 1 . (Case 1,4): W x and W y are both not monotone. The argument for (Case 1,3) can be adapted noting that d(x, u kmax ) ≈ σ k (x) max .
(Case 2,1): W y is monotone. There is a θ > 0 depending only on (P2) so that σ l+θ ≥ 3σ l for each l, and there is a C θ depending on (P2) so that σ l+θ ≤ C θ σ l for each l. Let l = ⌈lg d(x, u kmax )⌉ and fix w ∈ W y . If j(w) ≤ σ l+θ we have that any walk from x to w must pass through a socket point of order k max and so: ; as W y is monotone, d(w, u kmax ) ≥ j y (w) and so: and so d(w, x) j(w) where the constant in the lower bound can be estimated in terms of C 1 , C θ and θ.
(Case 2,2): W y is not monotone. We decompose W y as W (m) y ∪ {z 0 , · · · , z m = v y } and note that we can use (Case 2,1) on W y . For {z 0 , · · · , z m = v y } one can adapt the argument used in (Case 1,3).
(Case 3): π(x) ≤ π(y) ≤ π(u kmax ). This case can be dealt with along the lines of (Case 2) except in the case in which W y is not monotone, where a different estimate is required on the terminal part {z 0 , · · · , z m = v y }. Any walk from x to z i must pass through socket points of orders k max and k (y) max so that: but W y is not monotone, which implies σ k (y) max ≈ len W y which gives: 4. The exponents for which the Poincaré inequality holds 4.1. Geometric characterizations of the Poincaré inequality. The proof of the Poincaré inequality will involve the construction of families of curves joining points in G. Overall, we have preferred to avoid using the language of pencils of curves employed by [Sem96,Hei01], and preferred a probabilistic language. The rationale is that our construction is naturally modelled by Markov chains, a fact that also occurrs in the examples [CK15]. Specifically, we will deal with measurable functions defined on a probability space which take value in the set of (Lipschitz) curves on a metric space X; such maps will be called random curves. To a random curve Γ one can associate a measurable function defined on the same probability space and which takes values in the space of Radon measures on X by Γ → Γ (the length measure); such a map will be called a random measure. Finally, the maps to the end and starting points of Γ, Γ → end Γ and Γ → str Γ, produce random points in X. The support spt Γ of a random curve Γ is the set of edges that Γ crosses in positive measure with positive probability: To disprove the Poincaré inequality we will use the notion of modulus of families of curves, which we now recall.
Definition 4.2. Let P ≥ 1 and A be a family of locally rectifiable curves in the metric space X. We say that a Borel function g : X → [0, ∞] is admissible for A if for each γ ∈ A one has: Having fixed a background measure ν on X, we define the P -modulus of A, mod P (A), as the infimum of: where g ranges over the set of functions admissible for A. We will be mainly interested in modulus when A is the family A p,q of locally rectifiable curves connecting two points p, q, and when ν is of the form: where µ is a doubling measure on X and C > 0. In this case we will use the notation mod P (p, q; µ (C) p,q ) for the modulus of A p,q when the background measure is µ (C) p,q . We finally recall the definition of the Riesz potential centred on p: (4.6) µ p = d(p, ·) µ (B(p, d(p, ·))) µ.
The following Theorem summarizes a geometric characterization of (1, P )-Poincaré inequalities. It combines results of Heinonen-Koskela [HK98], Haj lasz-Koskela [HK95], Keith [Kei03], and Ambrosio, Di Marino and Savaré [ADS13], and the proof is included just for the sake of completeness. Note that we will take Theorem 4.7 as the working definition of the Poincaré inequality, and so we will not need to recall the usual definition of the Poincaré inequality.
Theorem 4.7. Let (X, µ) be a complete doubling metric measure space; then P ∈ I PI (X, µ) if and only if one of the following equivalent conditions holds: (1) There is a universal constant C such that for each pair of points p, q ∈ X one has: (4.8) d(p, q) P −1 mod P (p, q; µ (C) p,q ) ≥ C; (2) There is a universal constant C such that any pair of points p, q can be joined by a random curve Γ satisfying: ≤ Cd(p, q).
Consider the set A of locally rectifiable curves joining p to q; fix M large to be determined later and write A = A exit ∪ A long ∪ A good , where: (1) A exit consists of the locally rectifiable curves in A which meet X\B({p, q}, Cd(p, q)) in positive length; (2) A long are the locally rectifiable curves in A \ A exit which have length ≥ M d(x, y); (3) A good are the rectifiable curves in A \ (A exit ∪ A long ).
We will now fix µ (C) p,q as the background measure with respect to which we compute moduli; using the test functions g exit = 0 onB({p, q}, Cd(p, q)) and g exit = ∞ elsewhere, and g long = M d(p, q) onB({p, q}, Cd(p, q)) and 0 elsewhere, we see that: thus for M sufficiently large, (4.12) d(p, q) P −1 mod P (A good ) ≥ C/2.
Instead of computing modulus on A good we can compute it on the family of measures: (4.13) Applying the main result of [ADS13] we get a probability π on Σ good such that, denoting by ν = Σ good η dπ(η), we get: (4.14) dν = mod P (Σ good ) −1/P ; using (4.12) we conclude that: Now, to each η ∈ Σ good we can associate a unique unit-speed curve γ : [0, len γ] → X such that H 1 γ = η. Thus π becomes the law of a random curve Γ with E[ Γ ] = ν and then (4.9) follows from (4.15).
Take a random curve Γ satisfying (4.9) and let g be admissible for the curves joining p to q. Then: and (4.8) follows minimizing in g.

Construction of Random curves.
In this subsection we construct the ingredients to build the random curves used to verify the Poincaré inequality. This is the subsection where most of the technical work takes place. As we work with walks but need to produce random curves, we define the Lipschitz path associated to a walk as follows.
Definition 4.17. To a walk W = {w 0 e 1 w 1 · · · e l w l } we can canonically associate a 1-Lipschitz map Γ W : [0, len W ] → G by letting Γ W |[l, l + 1] be a unit speed parametrization of the edge e l .
We now define a notion of lift for walks used in the subsequent constructions.
Definition 4.18. Let W = {w 0 e 1 w 1 · · · e l w l } and w ′ 0 a point such that π(w ′ 0 ) = π(w 0 ). We construct a new walk {w ′ 0 e ′ 1 w ′ 1 · · · e ′ l w ′ l } as follows. The vertex w ′ i+1 is adjacent to w ′ i and is determined as follows. If w ′ i is not a socket point the requirement π(w ′ i+1 ) = π(w i+1 ) uniquely determines w ′ i+1 . Otherwise, assume that w ′ i is a socket point of order k and let e ′ i+1 denote the edge between w ′ i and w ′ i+1 . We require that λ(e ′ i+1 ; k) = λ(e i+1 ; k) and θ(e ′ i+1 ; k) = θ(e i+1 ; k) for all k. We say that W ′ is the lift of W starting at w ′ 0 and we will denote it by w ′ 0 · W . We now add some auxiliary definitions used in the constructions, e.g. when concatenating random curves.
Let p ∈ G a vertex and k ∈ N. We denote by F Θ (p, k) the set of those p ′ ∈ G satisfying π(p ′ ) = π(p), λ p ′ = λ p and θ p ′ (l) = θ p (l) for l > k. As above, to F Θ (p, k) we associate a canonical probability P by requiring: We now present the construction of a random curve which goes through a socket point ξ in G if one has a walk that passes through ξ. In the following, given a walk W = {w 0 e 1 w 1 · · · e l w l } we denote by W −1 the reversed walk {w l , e l , w l−1 , · · · , e 1 , w 0 }.
Proof. We prove (C1). Let ξ ′ = end(p ′ 0 · W 0 ); we use the notation w t , e t for the vertices, respectively the edges of W 0 ; we use the notation w ′ t , e ′ t for the corresponding edges and vertices of p ′ 0 · W 0 . We note that if t ≥ τ i + 1 (H3) implies that λ(e ′ t ; l) = {♠} for i ≤ l ≤ k − 1. We thus conclude that λ(ξ ′ ; l) = {♠} for l ≤ k − 1; for l ≥ k the label λ e ′ t coincides with that of λ et and so we conclude that λ(ξ ′ ; l) = λ(ξ; l) for l ≥ k. Therefore, ξ ′ is a socket point of order K. By (H2) all edges of W 0 have the same label θ, and this implies that all edges of p ′ 0 ·W 0 have the same label θ p ′ 0 . As π(ξ ′ ) = π(ξ), we conclude that ξ ′ is the point of F Θ (ξ; k − J cut ) with label θ p ′ 0 and thus (C1) follows. We now prove (C2). Note that the i-th vertices w i , w ′ i of W 0 and p ′ 0 · W 0 have π(w i ) = π(w ′ i ), and the labels (λ(w i ), θ(w i )), (λ(w ′ i ), θ(w ′ i )) and can differ only in the first k − J cut entries. Hence (C2) follows from Lemma 2.16.
We now prove (C3). First let e ∈ spt Γ and assume that e = e ′ l ∈ p ′ 0 · W 0 , e = e ′′ l ∈ p ′′ 0 · W 0 . As the path W 0 is monotone, l =l and there is a unique edge e s of W 0 such that π(e) = π(e s ). We can thus associate to e the unique integer in(e) = s. We now turn to the proof of (4.24). For p ′ 0 ∈ Λ(p 0 , k − J cut ) we will denote by e(p ′ 0 ; l) the l-th edge of p ′ 0 · W 0 . We now fix e ∈ spt Γ and assume that in(e) = s. We first consider the case s ∈ [1, τ k−1 ]. Then by (H5) there is a unique p ′ 0 ∈ F (p 0 ; k − J cut ) such that e is the s-th edge of p ′ 0 · W 0 . In this case by (H2)-(H3) lg(len W 0 − in(e)) is comparable to k up to a multiplicative constant depending on C 0 . Assume now that s ∈ (τ i , τ i+1 ]; then e is the s-the edge of p ′ 0 · W 0 if and only if: note also that in this case lg(len W 0 − in(e)) is comparable to i.
Note that in this case lg(len W 0 − in(e)) is comparable to 1. We can now put all this information together:  and so (4.24) follows by taking the quotient of (4.27) and (4.28).
Corollary 4.29. Suppose that W 0 satisfies the assumptions of Theorem 4.23 and let p ∈ G. Assume that for some C 0 > 0 one has: Then there is a C 1 = C 1 (C 0 , J cut ) such that: Proof. By assumption (4.30) we have that on the edges of spt Γ: (4.32) w ((λ(p; n), θ(p; n))) −1 .
We now obtain the following estimate using that W 0 |[τ i+1 , len W 0 ] has a number of edges σ i : w ((λ(p; n), θ(p; n)) −1 weight(µ; e) (4.33) In the following theorem we construct a random curve which moves "parallel" to a given walk W .
Theorem 4.34. Let W = {w 0 e 1 w 1 · · · e l w l } be a monotone walk joining p 0 to p 1 where ord(p i ) = 0. Let P i denote the canonical probability measure on F (p i ; k).
To each p ′ 0 ∈ F (p 0 ; k) we associate a walk W p ′ 0 as follows. We let w ′ 0 = p ′ 0 . Then, e ′ i and (hence) w ′ i+1 are determined by w ′ i and e ′ i−1 as follows. First π(e ′ i ) = π(e i ). If ord(w ′ i ) = 0 or w ′ i is not a gluing or a socket point the previous requirement uniquely determines e ′ i . If w ′ i is either a gluing or a socket point of order > k we take the edge e ′ i satisfying the additional requirement (λ(e ′ i ; ord(w ′ i ), θ(e ′ i ; ord(w ′ i ))) = (λ(e i ; ord(w i )), θ(e i ; ord(w i ))). If w ′ i is a socket point of order ≤ k then e ′ i is determined by the additional requirement that (λ e ′ i , θ e ′ i ) = (λ e ′ i−1 , θ e ′ i−1 ). Let Γ be the random curve determined by choosing p ′ 0 according to P 0 and letting Γ = W p ′ 0 . Then the following holds: (C1): end Γ has law P 1 ; (C2): spt Γ ⊂ B(Γ W , Cσ k ); (C3): For e ∈ spt Γ one has: where C 1 depends on (P1)-(P3) and Weight.
Proof. Fix p ′ 0 ∈ F (p 0 ; k) and let e t denote the t-th edge of W and e ′ t the t-th edge of W p ′ 0 . One has π(e t ) = π(e ′ t ); moreover, the choice of behaviour at gluing and socket points implies that: Thus, for e ∈ spt Γ there are a unique t ∈ N and a unique p ′ 0 ∈ F (p 0 ; k) such that e is the t-th edge of W p ′ 0 . We now prove (C1). Observe that the end point p ′ 1 of W p ′ 0 satisfies: Then, using the definition of the map τ in Definition 4.19, we get p ′ 1 = τ (p ′ 0 ) and so (C1) follows.
Statement (C2) is proven like in Theorem 4.23. We now show statement (C3). Let e ∈ spt Γ and let (t, p ′ 0 ) be the unique pair such that e is the t-th edge of W p ′ 0 . Then:  Assume that for some C 0 > 0 one has: and that len W ≤ C 0 σ k . Then there is a C 1 = C 1 (C 0 ) such that: Proof. By assumption (4.41) we have (w((λ(p; n), θ(p; n))) −1 .
on the edges of spt Γ. Then for e ∈ spt Γ one has: On the other hand, len W σ k and so: (4.45) In the following theorem we assume that the walk is monotone increasing for concreteness; the same result holds if the walk is monotone decreasing. The goal is to build a random curve which "expands" gaining access to new labels. This is needed to get the estimate (4.9).
Proof. We first explain why the construction of the walks W can be carried out. If C 0 is sufficiently large, one can ensure that whenever J cut ≥ C 0 , and if C is the constant appearing in Lemmas 2.9, 2.14, one has: We now explain why the concatenation in (4.54) is well-defined. Note that W ) −1 end at the point ξ ′ ∈ F Θ (ξ; k − J cut + 1) such that θ(p ′ 0 ; l) = θ(ξ ′ ; l) for l = k − J cut + 1. Therefore, the concatenation in (4.54) is well-defined.
We now turn to the proof of (C1). Let p ′ 0 = str Γ; conditional on the event E (old) one has that end Γ = p ′ 1 where p ′ 1 is the point of F (p 1 ; k − J cut + 1) satisying . The probability of the event: (4.61)
We thus conclude that (C1) holds For (C2) we can apply the same argument as in Theorem 4.23. We now prove (C3). The fact that in(e) is well-defined follows from the monotonicity of the walks W , W . As all edges of W have the same label, for p ′ 0 ∈ F (p 0 ; k − J cut ) one has that p ′ 0 · W = W p ′ 0 , where W p ′ 0 is defined as in Theorem 4.34. Therefore, the estimate (4.58) on the Radon-Nikodym derivative of E Γ (old) can be obtained from (4.35). Let now t ξ = in(e ξ ) where e ξ is the last edge of W (new) 0 . As remarked above, the walk W (new) 0 satisfies the assumptions of Theorem 4.23. Thus, if e ∈ spt Γ (new) and in(e) ≤ t ξ we can apply (4.24) to get (4.56) with T (e) = lg(len W 0 − in(e)). On the other hand, also the path (W (new) 1 ) −1 satisfies the assumptions of Theorem 4.23. In this case the point end Γ (new) avoids the sets of points p ′ 1 ∈ F (p 1 ; k − J cut + 1) such that: (4.65) (λ(p ′ 1 ; k − J cut + 1), θ(p ′ 1 ; k − J cut + 1)) = (s 0 , t 0 ); in applying Theorem 4.23 this can only introduce a multiplicative error lying in [(C gw,1 C gw,2 ) −1 , C gw,1 C gw,2 ] in the estimate (4.24). Note also that if in(e) ≥ t ξ , considering the reverse walk (W (new) 1 ) −1 , the integer in(e) in (4.24) must be replaced with len W − in(e) and thus the proof of (4.56) is complete.
Corollary 4.66. Let W be as in Theorem 4.46 and let p ∈ G. Assume that for some C 1 > 0 one has: Then there is a C 2 = C 2 (C 1 , J cut ) such that: Proof. We first apply convexity of the Q-th power of the L Q (µ p ) norm to get: Definition 4.70. Given P ≥ 1 we denote by Q the conjugate exponent P/(P − 1). Let I neck denote the range of exponents P ≥ 1 such that there is a C = C(P ) such that for each k ∈ N one has: As m k ≥ 2 one has that: (4.72) log 2 (w −1 {♠} C gw,1 ) + 1, ∞ ⊂ I neck ; on the other hand, if all m k are equal to some m one has: (4.73) Theorem 4.74. For P ∈ I neck the metric measure space (G, µ) satisfies a (1, P )-Poincaré inequality, i.e. I neck ⊂ I PI (G, µ).
Proof. We apply Theorem 4.7, i.e. for any pair of points (x, y) we show the existence of a random curve Γ satisfying: where C does not depend on x, y, and C Q does not depend on x, y but depends on Q. Γ is built by concatenating curves obtained by using Theorems 4.23, 4.34, 4.46.
We observe that if end Γ 0 = str Γ 1 the random curves Γ 0 , Γ 1 , up to translating their domains, can be concatenated to obtain a random curve Γ 0 * Γ 1 .
Step 1: First part of building ''half'' of a random curve joining x to y.
Fix points x, y and assume that max N(x, y) ≤ lg d(x, y). This assumption will be removed in Step 2. Using Theorem 3.6 we can choose a good walk from x to y satisfying (GWA1) and (GWA2). We let K = lg d(x, y). We thus have a uniform constant C 0 such that: For the moment let C be the maximum of the constants occurring at points (C2) of Theorems 4.23, 4.34, 4.46. We can find C 1 = C(C 0 ), J 1 = J(C 0 ) such that, if J ≥ J 1 andw satisfies: then one has: (4.80) d(w, x) ≥ C −1 1 i. We now subdivide W into subwalks {W α } α∈I (I is a finite set of integers), the idea being that W can be thought of as a concatenation of the {W α }. More precisely, this can be formalized by using a strictly increasing map α → m α , and letting W α denote the part of W starting at the m α -th vertex w mα and ending at the m α+1 -th vertex w mα+1 . Note that we obtain an order relation < on {W α } α∈I where W α < W α+1 .
Using the properties of the good walk constructed in Theorem 3.6 we obtain a J 2 such that there is a decomposition of W into monotone subwalks {W α } α∈I having the following properties: (Dec2): For each k ∈ N(x, y) such that θ x (k) = θ y (k), there is a W α = W (neck) k which can be decomposed into subwalksW 0 ,W 1 which satisfy the following: one has endW 0 = w s(k) = strW 1 ; moreover, for J cut ≥ J 2 the walksW 0 andW −1 1 satisfy the assumptions of Theorem 4.23 where ξ = w s(k) ; (Dec3): For each of the remaining walks W α there is a k such that: Γ is constructed by concatenating curves Γ α for each α ∈ I. This is done inductively, and one starts by letting Γ 1 = Γ W1 with probability 1. The next step depends on which of the conditions (Dec) is satisfied by W α+1 : • Case of (Dec1). We have W α+1 = W (exp) k and we know that end Γ α is a random point in F (w mα+1 ; k − J cut ) whose law is the canonical probability. We obtain Γ α+1 applying Theorem 4.46, so that end Γ α+1 is a random point in F (w mα+1 ; k − J cut + 1) whose law is the canonical probability. Moreover, by (4.81) we can apply Corollary 4.66 to conclude that: where C 2 is a uniform constant depending on the constants C 0 , C 1 , C, J 0 , J 1 , J cut . Moreover, by the assumption on P we have that there is a uniform constant C 3 depending on C 2 and Q such that: • Case of (Dec2). We have W α+1 = W (neck) k and we know that end Γ α is a random point in F Θ (w mα+1 ; k −J cut ) whose law is the canonical probability. We apply Theorem 4.23 to buildΓ 0 fromW 0 . We then take the canonical probability on F Θ (w mα+2 ; k − J cut ) and use again Theorem 4.23 to buildΓ 1 fromW −1 1 . We obtain Γ α+1 by concatenatingΓ 0 andΓ −1 1 subject to the following additional prescription; suppose that strΓ 0 = p ′ 0 ; then one takes is the canonical map of Definition 4.19. Note that: as the labels of the edges in sptΓ 0 and sptΓ 1 have different k-th entries. Moreover, as ξ = w s(k) and d(x, w s(k) ) ≈ C σ k , we can apply Corollary 4.29 to obtain the estimate: where C 2 is a uniform constant depending on the constants C 0 , C 1 , C, J 0 , J 1 , J cut . Moreover, by the assumption on P we have that there is a uniform constant C 3 depending on C 2 and Q such that: • Case of (Dec3). We know that end Γ α is a random point in F (w mα+1 ; k − J cut ) and that len W α+1 ≤ Cσ k . We build Γ α+1 by applying Theorem 4.34.
In particular, the assumptions of Corollary 4.40 are also met an so we have: where C 2 is a uniform constant depending on the constants C 0 , C 1 , C, J 0 and J 1 .
In this case W is given by Theorem 3.17. If θ x (k max ) = θ y (k max ) the construction can proceed as in Step 1 because at u kmax there is no change of the θ-label.
We now discuss the modifications for the case θ x (k max ) = θ y (k max ). We first enlarge W at w i = u kmax by inserting 4 subwalks {W i } 3 i=0 between w i and w i+1 . Let M = lg d(x, u kmax ), and let e denote the edge of W before u kmax . We takeW 0 to be a monotone geodesic walk whose edges have all the same label (λ e , θ e ), with lenW 0 = σ M and d(W 0 , x) ≥ C −1 1 σ M . ForW 1 we takeW −1 0 . Let now e denote the edge of W after u kmax . ThenW 2 is a monotone geodesic walk whose edges have all the same label (λ e , θ e ), with lenW 2 = σ M and d(W 2 , x) ≥ C −1 1 σ M . ForW 3 we takeW −1 2 . One then proceeds as in Step 1, by subdividing W . The subdivision must satisfy the additional requirement that the {W i } 3 i=0 are subwalks of the subdivision, and we have only to specify how to construct the corresponding {Γ i } 3 i=0 . OnW 0 we apply Theorem 4.34 and Corollary 4.40 and obtain the estimate: ThenΓ 1 andΓ 2 are built by applying Theorem 4.23 and Corollary 4.29 toW 1 and W −1 2 respectively. Note that strΓ 2 is taken to be a random point in F (strW −1 2 ; M − J cut ) whose law is the canonical probability. We buildΓ 12 by concatenatingΓ 1 and Γ −1 2 with the additional prescription that if strΓ 1 = p ′ 1 then strΓ 2 = τ (p ′ 1 ) where (4.93) τ : F (strW 1 ; M − J cut ) → F (strW −1 2 ; M − J cut ) is the canonical map of Definition 4.19. We thus obtain the estimate: Finally,Γ 3 is obtained by applying Theorem 4.34 and Corollary 4.40 toW 3 . We then have the estimate: With these modifications, one obtains (4.90), (4.91) where the constants have possibily worsened compared to Step 1.
Step 3: building a random curve satisfying (4.75), (4.76). Fix x, y ∈ G at distance > 1. We choose a vertex z of order 0 satisfying: we then choose J cut,x and J cut,y larger than J 2 of Step(s) 1, 2 such that: We then construct random curves Γ x connecting x to F (z; lg d(z, x)−J cut,x ), and Γ y connecting y to F (z; lg d(z, y) − J cut,y ) using Steps 1,2. Note that (4.100) implies that end Γ x and end Γ y have the same law. We can thus obtain Γ by concatenating Γ x and Γ −1 y . Now (4.75) follows from (4.91) and (4.96), (4.97). On the other hand, (4.76) follows from (4.90) and:  select two points p 0 , p 1 such that: (1) π(p 0 ) = m − σ k and π(p 1 ) = m + σ k ; (2) λ p0 = λ p1 ; (3) θ(p 0 ; j) = θ(p 1 ; j) if j = k + M and θ(p 0 ; k + M ) = θ(p 1 ; k + M ). Then there is a constant C 1 (C 0 , P ) such that: Proof. Let ξ ∈ B bad denote the socket point of label (λ, θ) such that π(ξ) = m. Let γ be a continuous curve joining p 0 to p 1 . Note that by possibly enlarging C 0 we have d(p 0 , p 1 ) ≈ C0 σ k and so for l(C 0 ) sufficienlty large, by Lemma 2.16 we have: If M (C 0 ) is sufficiently large, the only integer of order k + M contained in π(B bad ) is m. To estimate mod P (p 0 ; p 1 , µ p0,p1 ) we need to produce an appropriate Borel function g. For the moment we let g = ∞ on B c bad and then the case of interest becomes when γ stays in B bad ; in particular, γ must pass through a socket point ξ ′ ∈ F Θ (ξ; k + l).
Let s ∈ dom γ be the first time when γ(s) ∈ F Θ (ξ; k + l) and let γ 1 = γ|[0, s]. Note that: for i < k let t i = m − σ i and let ̺ i be the last time such that π • γ 1 (̺ i ) = t i . Let E(i) denote the set of edges e ∈ B bad such that π(e) ⊂ [t i , t i−1 ]. As there are no integers of order i in [t i , m] we conclude that the curve γ 1 |[̺ i , ̺ i−1 ] passes through edges {e 1 , · · · , e l } ⊂ E(i) such that: (E(i),1): e a and e a+1 are adjacent, t i ∈ π(e 1 ) and t i−1 ∈ π(e l ); 4): θ(e a ; j) = θ(p 0 ; j) for j > k + l. We now complete the definition of g by defining g|B bad as follows: if e ∈ E(i) for some i and (E(i),3) and (E(i),4) hold, we let g = (k − 1) −1 (σ i − σ i−1 ) −1 ; otherwise, we let g = 0. We now obtain the following lower bound: where we let σ 0 = 0. Note now that γ 1 |[̺ k−1 , s] is at distance ≈ C2 σ k from p 0 , p 1 , where C 2 is a uniform constant. Therefore, we have: (4.108) dµ note that g = 0 in B bad only on k−1 i=1 E(i), and letẼ(i) denote the set of edges of E(i) satisfying (E(i),3) and (E(i),4); as g vanishes on E(i) \Ẽ(i), we have for some C 1 (C 0 , C 2 , P, M, l): from which (4.104) follows.
Proof. We show that for any value of C, (1) in Theorem 4.7 fails. For any k ≥ 1 we can find a bad box B bad satisfying the assumptions of Lemma 4.102. Hence we find sequences of pairs of points (p As P ≤ 1 + log N (w −1 {♠} C gw,1 ) and as σ k /σ i ≤ N k−i , the rhs. of (4.111) goes to 0 as k ր ∞.
If all the m k are equal to some m, then the rhs. of (4.111) goes to 0 exactly when P ≤ 1 + log m (w −1 {♠} C gw,1 ).
Remark 4.112. Note that as w −1 {♠} C gw,1 ր ∞ one has min I PI (G, µ) → ∞, i.e. the range of exponents for which a Poincaré inequality holds gets narrower and narrower. On the other hand, as w −1 {♠} C gw,1 ց 1, min I PI (G, µ) → 1 and thus the range of exponents for which the Poincaré inequality holds can be arbitrarily prescribed. However, as either w −1 {♠} C gw,1 goes to 1 or ∞, the doubling constant of µ G blows up.

Stability under blow-up
In this section we show how to use G to construct a metric measure space X which satisfies the conclusion of Theorem 1.1. From now on the measure on G that we constructed in Section 2 will be denoted by µ G . In this section we often deal with balls of different spaces, and so at times we add a subscript to them to distinguish the space to which they belong. Given a metric space X, we use λX to denote X with the metric rescaled by the factor λ > 0. 5.1. Asymptotic cones. In this subsection we define asymptotic cones and construct the example X.
Definition 5.1. An asymptotic cone of a metric measure space (X, µ) is a measured pointed Gromov-Hausdorff limit of a sequence of rescalings: , p n where lim n→∞ λ n = ∞. Note that B X (p n , λ n ) denotes a ball of radius λ n in X, that is a ball of radius 1 in λ −1 n X. The set of asymptotic cones of (X, µ) will be denoted by as-Con(X, µ). Note that it would be more appropriate to say that as-Con(X, µ) is a set of equivalence classes of metric spaces under measure-preserving isometries, but we will avoid such subtleties in the following discussion.
Definition 5.3. A weak tangent (Y, ν, q) of a metric measure space (X, µ X ) is a measured pointed Gromov-Hausdorff limit of a sequence of rescalings: , p n where lim n→∞ λ n = ∞. The set of weak tangents of (X, µ X ) will be denoted by w-Tan(X, µ X ).
In the case of (G, µ G ) the fact that asymptotic cones exist and that the corresponding measures are doubling with uniformly bounded doubling constants follows from a standard compactness argument.

Proof.
On the metric level, the proof is straighforward using that one can approximate a weak tangent (Y, µ Y , q) ∈ w-Tan(X, µ X ) by rescaling an approximating sequence for (X, µ X , p). There is, however, an issue with normalization of balls which is addressed in the following lemma.
Lemma 5.6. Let (X, µ, p) ∈ as-Con(G, µ G ) and consider a sequence of rescalings: Then for each t ≥ 0 one has: Proof. Using that n → ν n (G) is lower semicontinuous if G is open and upper semicontinuous if G is compact, it suffices to show that one has, uniformly in p n , λ n : (5.9) µ G (B(p n , λ n t) \ B(p n , λ n (t − ε))) µ G (B(p n , λ n t)) ≤ O(ε 1/2 ).
We will use a discretization procedure of Gill and Lopez [GL14] that allows to compare PI spaces and graphs. We rephrase their result in a slightly more general context, where there is more freedom in the choice of the approximating graph; the proof is omitted being a straightforward generalization of their argument.
Theorem 5.13. Let H be a connected graph whose metric is a constant multiple of the length metric. For ε > 0 and C 0 > 0 consider a subset V of vertices of H which is an ε-separated net and C 0 ε-dense. Assume that for some C 1 > 0 there is a C 1 -biLipschitz embedding F : V → X such that F (V ) is C 1 ε-dense in X. Let µ X be a doubling measure on X with constant C 2 . Let µ H be a doubling measure on H which restricts to a multiple of arclength on each edge and such that one has, for some C 3 > 0: (5.14) µ H (B H (v, r)) ≈ C3 µ X (B X (F (v), r)) (∀(v, r) ∈ V × [ε, ∞)).
Since we work with pointed measured Gromov-Hausdorff convergence we need a local version of Theorem 5.13.
Corollary 5.16. In Theorem 5.13, assume that V is not C 0 ε-dense in the whole of H, but that V now lies in a ballB H (h, R) with R > 0 in which it is C 0 εdense. Assume also that F (V ) contains a C 1 ε-dense set in a ball B X (x, C −1 1 R). Furthermore, assume that X is geodesic. Then the conclusion of Theorem 5.13 holds replacing (H, µ H ) with: whereC depends only on C 0 , C 1 , C 2 , and ε.
Proof. One can reduce this local case to the global one, Theorem 5.13, by recalling that if (X, µ) is geodesic and admits a (1, P )-Poincaré inequality with exponent C(P ), there is a C 1 (C(P )) such that each for each R > 0 the metric measure space (B(x, R), µB(x, R)) admits a (1, P )-Poincaré inequality with constant C 1 (see [HK95]).
Step 1: I PI (X, µ X ) ⊂ I PI (G, µ G ). Let and assume that P ∈ I PI (X, µ X ), C(P ) being the corresponding constant. Choose N (n) such that: (5.21) 1 ≤ λ n σ N (n) ≤ N and pass to a subsequence such that lim n→∞ λn σ N (n) exists. Therefore, up to rescaling the metric on X by a factor in [1/N, 1] we can assume that: (5.22) (σ −1 N (n) G Gn , ν n , p n ) → (X, µ X , p); note also that (X, µ X ) is geodesic being a limit of geodesic metric spaces. Fix ε, R > 0; for n ≥ D 0 (R, ε) we can assume that the Gromov-Hausdorff distance between B Gn (p n , R) and B X (p, R) is at most ε 3 . Now the vertices of order ≥ l in G form a maximal σ l -net which becomes a maximal σ l σ −1 N (n) -net in G n ; for each n we choose N ε (n) ≤ N (n) such that: (5.23) ε ≤ σ Nε(n) σ −1 N (n) ≤ N ε. Lett V (n; ε) be the set of vertices of G n whose order in G is at least N ε (n) and which are contained in B Gn (p n , R). Then V (n; ε) is an ε-separated net in B Gn (p n , R) and is also N ε-dense there. Thus the cardinality of V (n; ε) is uniformly bounded in n and V (n; ε) → W in the Hausdorff sense where W is a 2 3 ε-separated net in B X (p, R) in which it is also 3 2 N ε-dense. Therefore for n ≥ D 0 (R, ε) we find an L-biLipschitz map: (5.24) F n : V (n; ε) → W, where L does not depend on ε or n. Now, as the cardinalities of V (n; ε) and W are uniformly bounded, for n ≥ D 1 (R, ε) we can assume that the sets V (n; ε) and W have the same cardinality and write V (n; ε) = {v (n) α } α∈A and W = {w α } α∈A so that F n (v (n) α ) = w α for each α ∈ A. We now use a variation on the argument of Lemma 5.6 (where we take balls not centred on the basepoints) to conclude that for each r ∈ [ε, R] one has: (5.25) ν n B Gn (v (n) α , R) → µ X (B X (w α , R)) ; so for n ≥ D 2 (R, ε) we can assume that: (5.26) ν n B Gn (v (n) α , R) ≈ 1+ε µ X (B X (w α , R)) .
For each s > 0 we can find n ≥ D 3 (s) such thatB G (p n , σ N (n) R/C cut ) contains an isometric copy B s ofB G (q, s) and such that the measures µ G B s and µ GBG (q, s) agree up to a multiple. Thus (5.29) B G (q, s), µ G B(G, q, s) admits a (1, P )-Poincaré inequality with constant C PI ; as C PI does not depend on s we conclude by letting s → ∞.
This follows from the stability of the Poincaré inequality under measured pointed Gromov-Hausdorff convergence, see [Kei03].

5.2.
Putting all together. In this subsection we complete the proof of Theorem 1.1.
Proof of Theorem 1.1. The existence of the measures {µ Pc } Pc follows combining Theorems 5.18, 4.74, 4.110 and Remark 4.112.
The projection map π : G → R passes to the limit giving a 1-Lipschitz map π : X → R. The geodesic lines of the form R × {λ} × {θ} pass to the limit and give a Fubini-like representation of the measure µ Pc . To this Fubini representation one can associate a Weaver derivation D, i.e. a horizontal vector field as in [Sch13].
The verification that (X, π) is a chart is standard and can be carried out in two ways. The first way uses a Sobolev-space argument like Sec. 9 in [CK15]. The second uses D and the Stone-Weierstrass Theorem for Lipschitz Algebras as in Example[Wea00, Example 5E].
The claim about the Assouad-Nagata dimension follows because the graph G has Assouad-Nagata dimension 1 and the Assouad-Nagata dimension is stable in passing to asymptotic cones.