Invariant embeddings of unimodular random planar graphs

Consider an ergodic unimodular random one-ended planar graph $\G$ of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By"locally finite"we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the Euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking, a unimodular embedding of $\G$ is one that is jointly unimodular with $\G$ when viewed as a decoration. We show that $\G$ has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the Euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings.

1. Introduction 1.1. Main results. Homogeneous (say, vertex-transitive) tilings of the Euclidean and hyperbolic planes are well-understood classical objects. Here we study random planar tilings that are "homogeneous" in some sense: they have an isometry-invariant law or satisfy a certain stationarity property, called unimodularity (see the definitions below).
We are asking the question: when does a random infinite graph that is known to be almost surely planar have such a "homogeneous" embedding into the Euclidean or hyperbolic plane without accumulation points (i.e., a locally finite embedding)?
After formalizing the above question, one finds right away that a necessary condition is that the random graph with a properly chosen root is unimodular itself, so from now on, we are only interested in this family of random rooted graphs. Unimodularity, embedding and representation by a tiling will be defined later in the introduction.
We say that the random rooted graph (G, o) has finite expected degree if the expected degree of o is finite. When G has an embedding into the Euclidean or hyperbolic plane, the relative location of the embedded vertices and edges by this embedding from the viewpoint of each vertex can be used to decorate the vertices. If the decorated graph is still unimodular, we call the embedding unimodular. The more precise definition is given in the next subsection.
Theorem 1.2. An ergodic unimodular random one-ended planar graph G of finite expected degree has a unimodular locally finite embedding • into the Euclidean plane if and only if G is invariantly amenable, • into the hyperbolic plane if G is invariantly nonamenable.
One can construct the embedding so that every edge is mapped into a broken line segment (piece-wise geodesic curve). We mention that here and in the next theorem, the "only if" part is missing from the second claim because of examples such as Example 1.6. With some extra condition this could be ruled out and have a full characterization in the theorems; see the discussion after the example. • in the hyperbolic plane if G is invariantly nonamenable.
The tilings guaranteed by the theorem are such that the expected area of the tile containing the origin is finite. Similarly to embeddings, we say that a tiling is locally finite if every bounded open subset of the plane intersects finitely many tiles. We will give precise definitions later in this section. We mention that the tiles in the above theorem can be required to be bounded polygons. Theorems 1.2 and 1.3 provide essentially complete dichotomic descriptions for the one-ended case. The cases not covered are those of G with 2 or infinitely many ends, in which situation a graph may or may not have any invariant embedding into one of the Euclidean or the hyperbolic plane (Remark 1.5). We do not treat 2 or infinitely many ends here, to avoid technical distractions from our main point.
In the case of invariantly amenable graphs, we will first construct an invariant embedding into the Euclidean plane, starting from a suitable invariant point process as the vertex set. This invariant embedded graph automatically defines a unimodular embedding. On the other hand, for invariantly nonamenable graphs, a unimodular embedding into the hyperbolic plane will be constructed directly, via circle packings.
(This circle packing embedding would not work in the Euclidean case, because Euclidean scalings provide an extra non-compact degree of freedom, making it unclear how to achieve unimodularity.) The intuitive claim that such a unimodular embedding is the "Palm version" of an invariant embedding does not seem to have been established in the hyperbolic setup; a similar statement in the Euclidean case is in the focus of [21]. This is the reason for the asymmetry in the Euclidean and hyperbolic cases of the above theorems. A new preprint by the second author and László Tóth [38] settles the question of invariant embedding of nonamenable graphs into the hyperbolic plane, together with the cases of 2 and infinitely many ended graphs, left open by the present paper.

1.2.
Definitions. Our focus is on Euclidean and hyperbolic spaces, hence the definitions will be phrased in this setting. One could ask questions in greater generality, for example by taking Lie groups as underlying spaces. We mention a few such directions in the concluding Section 6.
Without loss of generality from now on we assume that G is a simple graph, that is, it has no loop-edges or parallel edges. We also assume that all the degrees are finite in G.
Next we define unimodularity. First we are using random walks, as this seems to be more natural to describe when unimodular random embeddings are considered. Let G * be the collection of all locally finite connected rooted graphs up to rooted isomorphism, and let G * * be the collection of all locally finite connected graphs with a distinguished ordered pair of vertices up to isomorphism preserving this ordered pair. We often refer to an element of G * as a rooted graph (G, o), without explicitly saying that we mean the equivalence class that it represents in G * . Let (G, o) be a random rooted graph and suppose that o has finite expected degree. Reweight the distribution of (G, o) by the Radon-Nikodym derivative deg(o)/E(deg(o)). We will refer to such a reweighting by saying that we bias by the degree of the root. Denote the new random graph by (G , o ).
Let X 0 = o and let X 1 be a uniformly chosen neighbor of X 0 . We say that G = (G, o) is unimodular if (G , X 0 , X 1 ) has the same distribution as (G , X 1 , X 0 ). See [6] for the proof that this is equivalent to the original definition of unimodularity for graphs in [1], which we recall in the next paragraph. One may consider some decoration or marking on rooted graphs, and extend the above definition in the obvious way. Whenever there is a decoration, given as a function f on V (G) or as a subgraph U ≤ G, we denote this decorated rooted graph by (G, o; f ), (G, o; U ). In case of several decorations, we can list them all after the semicolon.
The original definition of unimodularity, equivalent to the previous one, is the following. Consider an arbitrary Borel function f : G * * → R + 0 . Then it has to satisfy the following equation (1.1) Here we do not distinguish between (G, x) as a rooted graph and as a representative of its equivalence class in G * * . This is standard in the literature and will not cause ambiguity. Equation (1.1) is usually referred to as the "Mass Transport Principle" (MTP). This equivalent definition of unimodularity naturally extends to decorated rooted graphs, one just has to consider Borel functions f from the suitable space.
Let M be some homogeneous metric space with some point 0 fixed; for our purpose we can just assume that it is a Euclidean or hyperbolic space. Let Isom(M ) be the group of isometries of M . For a graph G, an embedding ι of G into a Euclidean or hyperbolic space M is a map from V (G) ∪ E(G) that maps injectively every point in V (G) to a point of M , and every edge {x, y} to (the image of) a simple curve in M between ι(x) and ι(y), in a way that two such images can intersect only in endpoints that they share. The embeddings that we consider are locally finite, hence Let (G, o) be some unimodular random graph. For almost every G, let ι G = ι be some embedding of G into M . For every v, w ∈ V (G), assign the label dist(ι(v), ι(w)).
(In the literature it is more standard to assign labels to the vertices or edges, but assigning them to pairs of vertices is also a possibility, which is essentially equivalent to the other notion; see [5].) The embedded edges can also be encoded, as a label on each edge, coming from a suitable mark space in the space of continuous curves from [0, 1] to M up to orientation-preserving isometries of M . From this labelling, one can reconstruct ι up to Isom(M ). We say that ι is a unimodular embedding of G to M , if the labelling is a unimodular decoration of G. We emphasize again that a ι and any ι = γ • ι (γ ∈ Isom(M )) give rise to the same unimodular embedding.
A unimodular random graph (G, o) is invariantly amenable (or just amenable) if for every > 0 there is a random subset U ⊂ V (G) such that (G, o; U ) is unimodular, every component of G \ U is finite, and P(o ∈ U ) < . If this property fails to hold, then G is invariantly nonamenable (or just nonamenable). In the rest of this paper we will drop "invariantly" and simply call unimodular random graphs amenable or nonamenable, but the reader should keep in kind that these terms are different from the ones used for a deterministic graph. For the relationship of this notion of amenability to almost sure amenability or anchored amenability, see the discussion in [1] after the definition, and Theorem 8.5 therein for some equivalents. The notion of point-stationarity, introduced by Thorisson [32] for processes with a point at the origin, requires an invariance under rerooting, similarly to the random walk definition for unimodularity. In fact, the definition of point-stationarity for point processes applies to embedded graphs right away, and the existence of a unimodular embedding is equivalent to the existence of a point-stationary graph whose underlying graph has the same distribution as the given graph. Our observation (from [1]) that the Palm version of an isometry-invariant embedding is unimodular has been essentially known since Mecke [28], whose intrinsic characterization shows that the Palm version of a (translation-)invariant point process is always point-stationary.
We will denote the Euclidean plane by R 2 , and the hyperbolic plane by H 2 . Note that an invariant embedding of a unimodular graph automatically has finite intensity, the straight lines between neighboring ones (which will all have the same length), and apply the random isometry to this embedded graph of T 3 .
The next example shows that "zero intensity" is possible in case of unimodular embeddings.
Example 1.6. Consider G to be Z almost surely, and let M be the hyperbolic plane.
Take an infinite geodesic γ in M , fix a point 0 ∈ γ, and let g be an isometry of M that preserves γ and maps 0 to some g(0) = 0. Consider the embedded graph with vertex set {g i (0), i ∈ Z} and embedded edges being the pieces of γ between pairs g i (0) and g i+1 (0). One can check that this way we defined a unimodular embedding of Z into M . However, it is not possible to embed Z (or any amenable graph) into M in an isometry-invariant way.
One could define intensity for unimodular embeddings. We will not need this, but it could be defined as the reciprocal of the unique number r that ensures that the stable allocation on the embedded vertices with cell-volume-limit r is a full allocation (with r = ∞ standing for zero intensity); see [18] for the definition. Forbiding zero intensity, one could rule out pathologies as Example 1.6. Then Theorems 1.2 and 1.3 would become full characterizations (by a suitable modification of the proofs in Section 5 for the added "only if" parts).
A cyclic permutation of n elements is a permutation that consists of a single cycle of length n. A combinatorial embedding of a planar graph G is a collection of cyclic permutations π v of the edges incident to v over v ∈ V (G), and such that there is an embedding of G in the plane where the clockwise order of the edges on every vertex v {π v } is a unimodular decoration. The notion of unimodular combinatorial embeddings (and maps) was implicitly introduced in Example 9.6 of [1]. Note that this definition does not use any underlying metric on the plane, as it defines an embedding only up to homeomorphisms. For a given edge e, choose an orientation of e with e − being the tail and e + the head. Consider the edge π e − (e) oriented such that e − is its head.
Repeat this procedure for this new oriented edge, and iterate until we arrive back to (e − , e + ). Call the resulting sequence of edges a face of the combinatorial embedding.
One can check that the faces of actual embeddings coincide with the bounded domains surrounded by the respective faces of the corresponding combinatorial embedding.
In [36] it is shown that being unimodular and planar guarantees the existence of a unimodular combinatorial embedding in general, see Theorem 2.1. Call the resulting set the carrier of P . In the first case they called the graph CP parabolic, while in the second case they called it CP hyperbolic. They found several characterizing properties for this duality, such as the recurrence/transience of simple random walk. Earlier, Schramm [31] proved the uniqueness of these circle packings up to some transformations.
Theorem 1.8. (He-Schramm, [19], [20], Schramm, [31]) Let G be a one-ended infinite planar triangulated graph. Then G either has a circle packing representation whose carrier is the plane or it has a circle packing representation whose carrier is the unit disk, but not both.
• In the former case (when G is parabolic), the representation in the plane is unique up to isometries and dilations.
• In the latter case (when G is hyperbolic), the representation in the unit disk is unique up to Möbius transformations and reflections fixing the disk.
1.3. Connections to past research. Our topic is at the meeting point of isometryinvariant point processes and unimodular random graphs. The former has been a widely studied subject for many decades (see, e.g., the monographs [32], [25]), mostly in the setup of stationary (translation-invariant) point processes in Euclidean spaces.
A key problem in the present work is to represent certain graphs on the configuration points of a point process, in a covariant and measurable way. Questions of this flavor have been extensively studied for particular classes of graphs in the past, such as oneended trees, biinfinite paths ( [12], [17], [34]) or perfect matchings (see, e.g., [22], [16]).
Note however, that while in these settings only graphs on the vertex set have to be defined, in our context we also need to embed the edges into the underlying space.
Unimodular random graphs were first defined in [1], but similar ideas existed earlier (see references in [1]). They have attracted a lot of attention because of their connection to approximability by finite graphs (see [30] for the importance of such approximability in group theory), because of closely related notions in other areas (such as graphings in measurable group theory, see, e.g., [26]), and for being a natural generalization of group-invariant percolation. One can think of the notion of unimodular random graphs as a generalization of "percolation on a transitive graph with a unimodular group of automorphism, viewed from a fixed vertex", which makes it analogous to the Palm version of a point process. The direct connection is that invariant point processes as well as unimodular random graphs satisfy the Mass Transport Principle. That the Palm version of a random graph invariantly drawn in the plane is always unimodular as a planar graph was already proved by Aldous and Lyons [1] (see also our Remark 1.4), and our main results can be regarded as the converse to this claim.
A study of unimodular random planar graphs was initiated by Angel, Hutchcroft, Nachmias and Ray in [2] for the class of triangulations, and they showed that for a locally finite ergodic unimodular triangulated planar simple graph, being CP parabolic is equivalent to invariant amenability. In [3], unimodular planar graphs were further studied, without the assumption of being triangulated, but with the assumption that the unimodular graph comes together with a unimodular combinatorial embedding, in which case this joint object is called a unimodular planar map. Several criteria were identified as equivalents to invariant amenability. Theorems 1.2 and 1.3 can be thought of as further examples of the dichotomy.
As an example, consider the Uniform Infinite Planar Triangulation (UIPT), first defined by Angel and Schramm in [4]. The UIPT is a random graph that is unimodular (because it arises as the local limit of finite graphs) and planar (because all these finite graphs are planar), and moreover, the graph comes together with a unimodular com- Here we are interested in planar graphs, but it is reasonable to ask what happens in higher dimensions. In [35] it is shown (in the dual language of tilings) that every oneended amenable unimodular transitive graph has an isometry-invariant embedding into R d when d ≥ 3. The proof generalizes from transitive to random unimodular graphs. To our knowledge, the nonamenable (hyperbolic) case is open; Question 3.6 of [35] may be relevant in this regard.

Unimodular planar triangulation of unimodular planar graphs
The following theorem was proved in [36]. Recall that having a unimodular combinatorial embedding is a weaker requirement than having a unimodular or an invariant embedding, see Remark 1.7.  (G, o). and G + is a planar triangulation of finite expected degree. If G has one end then G + has one end.
Proof. By Theorem 2.1, G has a unimodular combinatorial embedding into the plane.
Fix such an embedding. The collection of faces is also jointly unimodular with G.
Let F be an unbounded face. By exploring the vertices of F along the boundary, a function from Z to the boundary is obtained which is not necessarily injective. Fix such a bijection, choose ξ ∈ {0, 1} uniformly at random, and for every pair {2k+ξ, 2k+ξ+1} (k ∈ Z), add a new vertex v k to the graph, and connect it to 2k + ξ, 2k + ξ − 1 and 2k + ξ + 1. Finally, add an edge between v k and v k+1 for every k ∈ Z. Now, in the resulting new graph we have a new infinite face, whose boundary is the biinfinite path  A bound p c < 1/2 is proved in [15] for general graphs with a minimum degree requirement.

Invariant circle packing representations of nonamenable graphs
The next theorem is a simple consequence of results by He and Schramm. One can turn the circle packing into a tiling of the same adjacency structure by properly subdividing every component of the complement of the disks into finite pieces and attaching them to suitably chosen neighboring disks. We omit the details.

Invariant embeddings of amenable unimodular planar graphs
In this section we will prove the amenable part of Theorem 1.1. We will later use this invariant embedding to constuct a unimodular embedding and prove the amenable   Some well studied planar unimodular amenable graphs are the uniform infinite planar triangulation (UIPT) ( [4]) and the augmented critical Galton-Watson tree conditioned to survive (AGW) (see, e.g., [27]).
Corollary 4.3. The AGW and the UIPT can be invariantly embedded in the Euclidean space.
We will need a special case of Theorem 5.1 from [35], illustrated on Figure 1.  Proof of Theorem 4.2. In the following constructions it will often happen that we need to choose some collection of pairwise inner-disjoint curves (broken line segments) connecting some given collection of pairs of points, within some specified bounded domain.
We will want to do it so that for a given isometry-invariant and measurable random collection of pairwise disjoint domains in R 2 the resulting collection of line segments over all domains is also invariant and measurable. Let us describe a method to do so. We will prove a stronger claim than the theorem, namely, that there exists an invariant embedding into R 2 that is consistent with Σ.
We may assume that G has uniformly bounded degree, as we explain next. We will introduce a new graph G , that we will obtain from G by replacing vertices by paths, with two such path adjacent if and only if the original vertices were, and in a way that all degrees in G will be at most 3. The construction will give rise to a combinatorial embedding Σ of G . Then we will show that an invariant embedding of G consistent with Σ can be used to define one for G that is consistent with Σ. The simple construction is summarized on Figure 2. and L(v(i), w(j)), where n(v, i) = w and n(w, j) = v. We end up with an embedding of G from the embedding of G as desired; see Figure 2. So from now on we will assume that G has bounded degrees. If x, y ∈ L k , x = y, then T x and T y are disjoint. Let G i be the graph on vertex set V (G) and edge set Then G i is a unimodular finite exhaustion of G. Finally, define γ(v) = τ (v) whenever v ∈ L 0 , and let γ(v) be the ball of radius dist(v, ∂τ (v))/2 around v when v ∈ L 0 . Let Γ(v) be the interior of the closure of ∪ u∈Tv τ (u).
will embed the edges of K v in Γ(v), and the above facts guarantee that no two points of ι(V (G \ K v )) are separated in R 2 by these embedded edges. Furthermore, for all edges not in K v but incident to K v , we will define an embedded half-edge, connecting the endpoint to ∂Γ(v). The procedure is illustrated on Figure 3.
We will proceed in steps, embedding G n in step n, and also embedding the half-edges coming from G \ G n . This will be done in a way that (1) the embedding of G n is an extension of that of G n−1 . In particular, for every e ∈ G n \ G n−1 the embedded image of e contains its embedded half-edges from the previous step, and for every edge of G n−1 the embedding in step n is the same as in step n − 1.
(2) All edges {v, w} with w ∈ K v , v ∈ L n , are embedded in the interior of the closure of τ (v) ∪ Γ(w); (3) similarly, if an edge in E(K v ) \ G n−1 has its endpoints in the distinct subtrees T w and T w of T v , with w and w adjacent to v ∈ L n , then the edge is embedded into the interior of the closure of τ (v) ∪ Γ(w) ∪ Γ(w ).
(4) All half-edges starting from a vertex in K v , v ∈ L n , reach ∂Γ(v).
As a preparatory step (step 0), consider G 0 , the empty graph on V (G). For every v ∈ V (G), pick some embedding of the half-edges starting from ι(v) ∈ γ(v) to randomly chosen points of the boundary of γ(v), in a way that the embedding of the halfedges is consistent with Σ. The embedding defined for step 0 trivially satisfies (1)-(4).
It is consistent with Σ by definition, and any extension will also be consistent with Σ. We proceed to step n recursively. For every component . Then Γ(v) contains the embedded points of V (K), moreover, the Γ(v i ) are pairwise disjoint open subsets of Γ(v) containing the embedded K v i respectively.
Contract every K v i in G to a single point, and also contract G\K v (which is a connected graph) to a single point x ∞ . The resulting finite graph K inherits a combinatorial embedding from Σ. Consider an arbitrary embedding of K to the 2-sphere that represents this combinatorial embedding. Remove an infinitesimally small neigborhood of the embedded nodes, to get a 2-sphere with holes in it and pairwise disjoint arcs connecting some of these holes. It is easy to check that there is a homeomorhism from this surface to τ (v), which maps the boundary of the hole belonging to x ∞ to ∂Γ(v) ⊂ ∂τ (v), and maps the boundary of the hole corresponding to the contracted . Furthermore, this homeomorphism can be chosen so that the images of the drawn segments are continuations of the respective drawn half-edges in the Γ(v i ). We have just shown that there exists an embedding of the component K v of G n into Γ(v) that satisfies (1)-(4); now choose the broken line segments for one such embedding randomly in a way as described at the beginning of this proof.
Since every step is an extension of the previous one, we obtain an embedding of G in the limit, as desired. Local finiteness is guaranteed by the fact that every τ (u) (u ∈ V (G)) is intersected by finitely many embedded edges. In [35] it is proved that every amenable unimodular transitive graph can be represented by an invariant tiling of R d for d ≥ 3, and the proof extends from transitive to random right away.
Proof. Consider the embedding of G into R 2 as in Theorem 4.2 and let P be the point process that the embedded vertices define. To each face F and point v ∈ ∂F , v ∈ P, we will assign a piece of the face incident to v, in such a way that two such pieces share a 1-dimensional boundary iff the corresponding vertices are adjacent. For the case of bounded faces one can apply a modified "barycentric subdivision", see Figure 4: for each pair v and w of adjacent vertices that are consecutive along F , consider the broken line segment representing the edge between them, and consider its midpoint, that is, the point that halves the length of the broken line. Choose some point uniformly in F , and connect it to all these midpoints by some broken line. If F is infinite, we will apply a trick similar to the one in [24]. For every pair v and w of adjacent vertices such that ι(v) and ι(w) are consecutive along F , let h(v, w) = h(w, v) be the midpoint of the broken line segment between them. Choose a conformal map f between F and the upper half plane H of C that maps infinity to infinity. By the standard extension of f −1 to the boundary ∂H, we can define a set of f -images in R for every h(v, w) ∈ ∂F .
(This set consists of one or two points, depending on whether the broken line between v and w has F on only one side or on both.) Let a ∈ R be one such image, and consider the vertical line L a = {a + bi : b ∈ R + } in H. Consider f −1 (L a ) for all the a. One can check that they subdivide F into pieces as we wanted. It is also clear that the construction does not depend on the choice of f (which is unique up to conformal automorphisms of the upper half plane of the form x → ax + b, a, b ∈ R, a = 0), and that it is invariant. See [24] for a detailed argument. It remains to prove that the tile of the origin has finite area almost surely. This is known for any invariant point process in R 2 (whose intensity is automatically positive) and partition as in our setup, see, e.g., (9.15) in [10].
We expect that with some extra work one could also get a tiling where every tile has area 1. One would have to ensure that the embedded vertices in Theorem 4.2 form a point process where the number of points in large boxes is relatively close to the expectation, and then build up the tiling stepwise and directly, eventually assigning unit tiles to all vertices. We have not worked out the details.
What seems to be harder to control, is the diameter of the tiles. Various invariance principles follow from [14] if one is able to construct an initial embedding for the given graph that satisfies a certain finite energy condition. The embeddings are assumed to be translation invariant modulo scaling. Whether our method can be useful in this setting is to be investigated in the future.

Proofs of the main theorems
Proof of Theorem 1.1. The existence of such representations if G is amenable is proved in Theorems 4.2 and 4.5.
For the "only if" part, suppose first that a nonamenable G had an isometry-invariant embedding as in Theorem 1.2 into R 2 . Then one could use the invariant random partitions of R 2 to 2 n times 2 n squares to define a unimodular finite exhaustion of G.
Thus G has to be amenable, a contradiction.
Proof of Theorem 1.2. For the nonamenable case the unimodular embedding into H 2 is given in Theorem 3.1. That there is no such an embedding into R 2 is proved the same way as in the proof of Theorem 1.1.
If G is amenable, an isometry-invariant embedding into R 2 exists. With the same arguments as in Example 9.5 of Aldous and Lyons in [1], the Palm version of this random embedded graph (as a graph with the decoration given by the embedding and rooted in the origin) is unimodular.
Proof of Theorem 1.3. The "if" parts of the claims follow from Theorem 4.5 and 3.1.
(Although Theorem 3.1 is for simple graphs, the tiling obtained from a circle-packing can be extended when there are parallel edges or loops.) For the "only if" part, note that an invariant tiling gives rise to an invariant embedding (choose a uniform random point in each tile and suitably connect it to its neighbors). Hence the claim is reduced to that in Theorem 1.2.

Further directions and open problems
To conclude, we propose a number of questions that are more or less connected to our main topic.
A natural direction could be the following. Say that two unimodular Lie groups M and N are equivalent, if every, possibly random, tiling T that invariantly tiles M with compact tiles also invariantly tiles N . Here tiles are compact simply connected.
Assume there is a tiling that invariantly tiles both M and N .
• Must they be quasi-isometric? We found that this is not necessarily the case.
As proved in [35], R d can be invariantly tiled with bounded tiles by any amenable transitive one-ended graph, whenever d ≥ 3. In particular, both R 3 and R 4 can be invariantly tiled by Z 3 , yet they are not quasi-isomorphic.
• Must they be equivalent? Not necessarily. Z 2 invariantly tiles R 2 , and also R 3 (by the just mentioned result). But the two are not equivalent, because no nonplanar graph (such as Z 3 ) can tile R 2 .
Bonk and Schramm [9] constructed a quasi-isometric embedding of hyperbolic graphs into real hyperbolic spaces.