On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that ‘may be made isometric’ is not a transitive relation.


List of groups of dimension at most three
Many of these groups are well known: the k-dimensional Euclidean group R k , the kdimensional torus T k = (R/Z) k and direct products of these groups. Nilpotent but non-Abelian groups are the Heisenberg group N 3 (R) and its quotient N * 3 (R) modulo the group of integer points in the center, when N 3 (R) is seen as upper triangular matrix group. Among the solvable but not nilpotent groups there are Aff + (R) (the group of orientationpreserving affine maps of the real line) and products thereof with R and T 1 , as well as SE (2) [the universal cover of the group SE(2) of orientation preserving isometries of the Euclidean plane] and SE(2) k [the k-fold cover of SE (2)]. Well-known simple groups are SU(2) (the special unitary group), SO(3) (the special orthogonal group), SL(2) (the universal cover of the special linear group), and PSL(2) k [the k-fold cover of the projective special linear group PSL (2)].
Apart from SL (2) and SU (2), all the simply connected groups listed in the previous paragraph are isomorphic to semidirect products R 2 A R, where R acts on R 2 by a matrix A ∈ Mat(2 × 2, R) such that the Lie group product is given by the following expression: One can find a basis {E 1 , E 2 , E 3 } for the Lie algebra of R 2 A R whose structure constants are given by we obtain J (respectively D λ , respectively C λ ).

Classification results
Standing assumption. All distances considered are left-invariant Riemannian distances. A (not necessarily continuous) map : (X , d) → (X , d ) between metric spaces is a quasi-isometry if there exist constants 0 ≤ C < ∞ and 1 ≤ L < ∞ such that (i) L −1 d(x, y) − C ≤ d ( (x), (y)) ≤ Ld(x, y) + C for all x, y ∈ X , (ii) for all x ∈ X there is x ∈ X such that d ( (x), x ) ≤ C.
If (i) and (ii) hold with C = 0, the map is said to be bi-Lipschitz, and if moreover L = 1, then is an isometry. If X and X are manifolds and if the distances d and d are induced by Riemannian metrics g and g, respectively, then according to a well-known result by Myers and Steenrod [28], the map is an isometry exactly if it is a diffeomorphism such that * g = g, see also [32,Theorem 5.6.15]. Since any two left-invariant Riemannian distances on a Lie group are bi-Lipschitz equivalent, we can discuss the quasi-isometric and bi-Lipschitz classification of such groups without specifying a metric. On the other hand, the existence of isometries between two groups depends on the choice of metrics. As we are interested in the geometric classification of groups, rather than the classification of groups endowed with a specific metric, we study the following property. If X is a fixed model space with a standard distance d X , for instance Euclidean space or the hyperbolic plane, we will also say that "G may be made isometric to X " if there exists a left-invariant Riemannian distance d G on G such that (G, d G ) and (X , d X ) are isometric.
In Sect. 2, we discuss relations between connected Lie groups of dimension at most three in descending order of strength, that is, we list pairs consisting of groups that (a) may be made isometric (Proposition 2.2) (b) are bi-Lipschitz (Proposition 2.11) (c) are quasi-isometrically homeomorphic (Proposition 2.14) (d) are quasi-isometric (Proposition 2.15).
To conclude the quasi-isometric classification given in Theorem 1.2 below, we show that the pairs not appearing in the list (a)-(d) consist of groups that are not quasi-isometrically equivalent.
Classification problems for Lie groups have a long history that dates back to Bianchi's [2] isomorphic classification of 3-dimensional Lie algebras. This note is concerned with the geometric classification of Lie groups that are additionally equipped with left-invariant Riemannian distances. Gromov [14] in his address to the ICM in 1983 promoted a program to study finitely generated groups with word metrics up to quasi-isometries. This classification problem is related to the quasi-isometric classification of Riemannian manifolds, as the fundamental group of a compact connected Riemannian manifold M is a finitely generated group quasi-isometrically equivalent to the universal Riemannian cover M according to the Švarc-Milnor lemma.
In the first part of this note, we recall the quasi-isometric classification of connected Lie groups up to dimension three. This is the work of several authors who have studied various aspects of the quasi-isometric classification, for instance for solvable groups of a specific form, or under curvature constraints. We list some of these results: Guivarc'h and Jenkins' [17,19] characterization of connected Lie groups with polynomial growth, Heintze's [18] work on solvable Lie groups and homogeneous manifolds of negative curvature, Milnor's [27] study of the curvature properties of left-invariant Riemannian metrics on Lie groups, the study of 3-dimensional model geometries and Dehn functions in the work of Epstein et al. [8] on automatic group, Pansu's [30,31] work on L p cohomology, de Cornulier's [6] computation of the covering dimension of asymptotic cones of connected Lie groups, the study of quasi-isometries of certain solvable Lie groups by Eskin et al. [9], and Xie's [36] quasi-isometric classification of negatively curved solvable Lie groups of the form R n R. Depending on the case to be treated, different tools are used in the classification problem, such as volume growth, Dehn functions, curvature and asymptotic cones of Riemannian manifolds.

Theorem 1.2 (Various authors) All connected real Lie groups of dimension at most three are classified up to quasi-isometries according to the following table:
Class Representatives We stress that the classes (7 λ ) are distinct for different values of λ, and the same holds for (11 λ ). In Sect. 3 we will explain how the above mentioned results by various authors can be combined to prove Theorem 1.2.
According to Theorem 1.2, two simply connected 3-dimensional Lie groups G and H (that are not isomorphic) are quasi-isometric to each other if and only if one of the following holds: In Proposition 2.2 we shall show that in all these cases, the two groups G and H may in fact be made isometric. By Proposition 2.1, this means that there exists a Riemannian manifold M on which both G and H act simply transitively by isometries. In fact, M may be taken equal to a Riemannian manifold that corresponds to one of the eight 3-dimensional model geometries by Thurston [35]: • the Euclidean geometry in (1), • the geometry of SL(2) in (2), • the hyperbolic geometry in (3), see the discussion in Sect. 2.1, and in particular Remark 2.8 for (2). Thus we obtain the following result.

Theorem 1.3
If two non-isomorphic simply connected 3-dimensional Lie groups are quasiisometric, then they may be made isometric to one of the eight Thurston geometries.
In Proposition 2.11 we shall show that without the assumption "simply connected", it is not true in general that two connected, quasi-isometric Lie groups may be made isometric. Moreover, since the groups PSL(2) k , for different values of k ∈ N, may all be made isometric to Aff + (R) × T 1 , but cannot be made isometric to each other, we have the following consequence.

Proposition 1.4
The relation "may be made isometric" is not transitive.

Groups that may be made isometric
We begin the section with a basic observation about Lie groups that may be made isometric and carry on with a list of 3-dimensional Lie groups that may be made isometric.

Proposition 2.1 Two Lie groups G and H may be made isometric if and only if there exists a Riemannian manifold M on which both G and H act simply transitively by isometries.
Proof Assume first that G and H possess Riemannian distances d G and d H , respectively, for which there exists an isometry : (G, d G ) → (H , d H ). Take M = H equipped with the Riemannian metric g that induces d H . Clearly, H acts on M simply transitively by isometries, and the same is true for G with the action given by where L g denotes left translation by g ∈ G.
Conversely, assume that G and H act simply transitively on a manifold M with Riemannian distance d. Fix x 0 ∈ M and define Since by assumption the actions of G and H on M are free, the above definition yields distance functions on G and H . From the compatibility of group actions and the fact that G and H act by isometries, we easily deduce that d G and d H are left-invariant. For instance, for G, we find for Since the given actions by G and H on M are also transitive, for every g ∈ G we find h(g) ∈ H such that g.
which is easily seen to be an isometry.

Proposition 2.2
Each of the following pairs consists of groups that may be made isometric: Proof It is well known that R 3 and SE(2) may be made isometric, see for instance [27,Corollary 4.8], [26, Theorem 2.14, (1-b)], and [23, §4]; or read the discussion later in this section. The statement that SE(2) k may be made isometric to R 2 × T 1 is Proposition 2.3. As a corollary, the groups SE(2) k and SE(2) k for arbitrary k, k ∈ N may be made isometric. Proposition 2.5 shows that SL(2) and Aff + (R) × R may be made isometric. By Proposition 2.10, Aff + (R) × T 1 may be made isometric to PSL(2) k for every value of k ∈ N.
The items (6) and (7) in Proposition 2.2 follow by curvature considerations. On the (simply connected) groups D 1 and on C λ , λ > 0, one can find a left-invariant Riemannian distance with constant negative sectional curvature: for D 1 , this follows from Special Example 1.7 in Milnor's article [27], for C λ , λ > 0, it is a consequence of [27,Theorem 4.11]; see also [26, Lemma 2.13 and Theorem 2.14, (1-a)] and [36,Introduction]. It is well known that every simply connected and complete Riemannian manifold with negative constant sectional curvature K is isometric to hyperbolic space in the respective dimension with sectional curvature K , hence all the groups D 1 and C λ , λ > 0 may be made isometric to hyperbolic 3-space, and thus also to each other.
We now provide the details for the results that have been used in the proof of Proposition 2.2 and for which no other reference has been given. The groups to be considered are SE(2), SL (2), and quotients thereof. The simply connected Lie group SE (2) A direct computation shows that the Euclidean distance d E on R 3 is left-invariant with respect to * , and hence R 3 and SE(2) may be made isometric. It is easy to verify that the sets (N k , * ), k ∈ N, given by are exactly the discrete normal subgroups of SE(2). Every k ∈ N gives thus rise to a multiply connected Lie group The center of SE(2) k contains exactly k elements, which shows that SE(2) k is not isomorphic

Proposition 2.3
For every k ∈ N, the group SE(2) k may be made isometric to the standard round cylinder R 2 × R/Z.

Proof
We construct a left-invariant distance on SE(2) k , by setting provides an isometry between R 2 × R/Z and SE(2) k .
We now turn our attention to SL(2) and its quotients. Since SL(2) is a simple Lie group, [5, Corollary 3.11] is useful.

Theorem 2.4 (Cowling et al.) Let G be a connected semisimple Lie group and let G = AN K be its Iwasawa decomposition. Write K as V × K , where V is a vector group and K is compact. Then G may be made isometric to the direct product AN
If K is compact, then G may be made isometric to AN × K . A condition which ensures the compactness of K for a given semisimple Lie group is that G has finite center, see [12, p.160 in Chapter 4]. A connected semisimple Lie group that is linear has finite center, see for instance [12,Chapter 1,§5].
The Iwasawa decomposition of SL (2) is AN K , where A and N are the following matrix groups and K is isomorphic to R. More precisely, the Iwasawa decomposition is given by the diffeomorphism where π : SL(2) → SL(2) is the universal covering projection. Note that AN is isomorphic to the orientation-preserving affine maps of the real line, that is, to Aff + (R). Theorem 2.4 applied to the Iwasawa decomposition of SL(2) yields the following statement.

Proposition 2.5
The groups SL(2) and Aff + (R) × R may be made isometric.

Remark 2.6
The group Aff + (R) admits a left-invariant metric of constant negative sectional curvature (see for instance [27,Special Example 1.7]) and hence, by the same reasoning as in the proof of Proposition 2.2, it may be made isometric to the hyperbolic plane H 2 . The quasi-isometric, or even bi-Lipschitz, equivalence of H 2 × R and SL(2) was proved earlier by Rieffel [33] in her Ph.D. thesis. The idea of the construction is explained in [22, §2]. To set the stage, we follow [34, p. 462] and observe that the standard Riemannian metric on H 2 induces a natural Riemannian metric on T H 2 in such a way that for every isometry f : H 2 → H 2 , the differential d f : T H 2 → T H 2 is an isometry as well. Since the unit tangent bundle U T H 2 is a submanifold of T H 2 , it inherits a Riemannian metric from T H 2 , and as U T (H 2 ) may be identified with PSL(2), this metric lifts to SL (2). One can show that the thus obtained Riemannian metric on SL(2) is left-invariant, see [34, p. 464].
To prove the bi-Lipschitz equivalence of H 2 × R and SL (2), one constructs a map Since the groups SL(2) and Aff + (R) × R may be made isometric, one might wonder if there is a "standard" Riemannian manifold to which they may both be made isometric. According to Remark 2.7, this manifold cannot be the standard H 2 × R, but it turns out that SL(2) endowed with the metric corresponding to one of the Thurston geometries has the desired property; see Remark 2.8 below.
Consider the left-invariant Riemannian metric g SL(2) on X := SL(2) that arises from the identification of PSL(2) with the unit tangent bundle U T (H 2 ) as described in Remark 2.6 and let G := Isome( SL(2)) be the corresponding isometry group. Then (X , G) is one of the eight three-dimensional model geometries of Thurston [35,Theorem 3.8.4]. Clearly, SL(2) acts transitively by isometries on (X , g SL (2) ). The following remark shows that the same is true for Aff + (R) × R. According to Proposition 2.1, this also provides another proof for Proposition 2.5.

Remark 2.8
The group Aff + (R) × R acts simply transitively by isometries on X endowed with the Riemannian metric that corresponds to Thurston's model geometry on SL (2). To see this, consider the group G := Isome( SL(2)), which has been discussed in [34, p. 464 ff].
It has been shown that G consists of two components, say and . The identity component is a 4-dimensional Lie group generated by the actions of R and SL(2) on X . The action of SL(2) is immediate, and according to the Iwasawa decomposition, it yields in particular an action of Aff + (R) on X . To explain the action of R, it is useful to see X as a line bundle over H 2 . The center of SL (2), which is isomorphic to the additive group Z, acts on X by preserving the line bundle structure and covering the identity map of H 2 . This action extends to an action of R on X by translation of the vertical fibers [this action arises as an action of S 1 on U T (H 2 ) which covers the identity of H 2 and rotates each fibre by the same angle]. Since the action of R commutes with the action of SL (2) [and thus of Aff + (R)], we obtain that Aff + (R) × R acts by isometries on X . Moreover, since Aff + (R) × R acts transitively on the base manifold H 2 of X , and R acts by translation on the vertical fibers, we see that Aff + (R) × R acts transitively on X . Finally, we argue that the action is free. Assume that (g, s).x = x for some g ∈ Aff + (R), s ∈ R and x ∈ X . Then, since the action of R covers the identity map of H 2 , it follows that g.x and x must lie in the same vertical fibre of X . As the action of Aff + (R) on X is induced by a free action of Aff + (R) on H 2 , it follows that g = e, as desired. Moreover, s = 0 since the action of R is free. This shows that Aff + (R) × R acts simply transitively by isometries on (X , g SL (2) ).

Remark 2.9
As the classification in Theorem 1.2 shows, already in dimension 3 the property of admitting a lattice (i.e., a discrete subgroup of cofinite volume) is not a quasi-isometric invariant. For example, the group Aff + (R) × T 1 is not unimodular by [27, Lemma 6.3] and hence cannot have lattices (see [27,Section 6] or [1,Proposition 2.4.2]), yet it is quasiisometrically equivalent to SL(2) = SL(2, R), which admits the lattice SL(2, Z).
For k ∈ N, the Iwasawa decomposition of PSL(2) k is where K k is the k-fold cover of the projective special orthogonal group PSO(2). Theorem 2.4 applied to the Iwasawa decomposition of PSL(2) k yields the following result.

Proposition 2.10
For every k ∈ N, the group PSL(2) k may be made isometric to Aff + (R) × T 1 .

Proposition 2.11
The groups PSL(2) k and PSL(2) k for different values of k, k ∈ N are bi-Lipschitz equivalent, but cannot be made isometric.
The bi-Lipschitz equivalence of PSL(2) k and PSL(2) k follows easily from Proposition 2.10, but to show that these groups cannot be made isometric, we use [13, Theorem 2.2] by Gordon, which we restate here for the reader's convenience.
Assume that A is a connected Lie group with a connected subgroup G. Choose Levi factors G s and A s of G and A, respectively, such that G s ⊂ A s , and denote by g s and a s the Lie algebras of G s and A s . By definition, the Lie algebras g s and a s are semisimple and thus a direct sum of simple Lie algebras, some of which may be compact and others not. This leads to the direct sum decomposition where g c is the direct sum of all compact simple ideals of g s and g nc is the direct sum of the remaining simple ideals. In the same way, one decomposes a s = a nc ⊕ a c . By G nc and A nc we denote the connected subgroups of A with Lie algebras g nc and a nc , respectively.

Theorem 2.12 (Gordon) Assume that A is a connected Lie group with a connected subgroup G whose radical is nilpotent. Suppose further that there exists a compact subgroup K of A such that A = G K . Then A nc = G nc .
With this theorem at hand, we can prove Proposition 2.11.
Proof of Proposition 2.11 By Proposition 2.10, both PSL(2) k and PSL(2) k may be made isometric to Aff + (R) × T 1 . Thus there exist left-invariant Riemannian distances, say d k and d k on Aff + (R) × T 1 , as well as d on PSL(2) k and d on PSL(2) k such that (PSL (2) Since d k and d k are bi-Lipschitz equivalent, it follows that PSL(2) k and PSL(2) k are bi-Lipschitz equivalent.
Next we show that PSL(2) k and PSL(2) k cannot be made isometric. For k ∈ N, we fix a left-invariant Riemannian distance d G on G := PSL(2) k and we let A be the isometry group of (G, d G ). Then A = G K as in Theorem 2.12, with K = Stab(e) ∩ A, where Stab(e) denotes the stabilizer of the identity in G. Since G is simple, its radical is trivial and hence nilpotent and moreover, G nc = G. It follows by Theorem 2.12 that G = G nc = A nc . The same reasoning applies for k instead of k, so that we obtain G = A nc for G = PSL(2) k and A the isometry group of (G , d G ). Now if (G, d G ) and (G , d G ) were isometric, then A would be isomorphic to A with an isomorphism given by conjugation via the isometry between (G, d G ) and (G , d G ). This would imply that PSL(2) k = A nc is isomorphic to A nc = PSL(2) k , which is possible only if k = k [otherwise the centers of PSL(2) k and PSL(2) k have different cardinality and hence the groups cannot be isomorphic].

Quasi-isometrically homeomorphic groups
We now consider multiply connected groups that are homeomorphic via a quasi-isometry but not bi-Lipschitz equivalent. The latter fact will be proved by contradiction: if there existed a bi-Lipschitz homeomorphism between the groups it would lift to a bi-Lipschitz homeomorphism of the universal covers according to Proposition 2.13. We first recall some basics from covering theory.
Assume that G is a simply connected Lie group equipped with a left-invariant Riemannian metric g. If N is a discrete normal subgroup of G, then G/N is a connected Lie group which admits a unique left-invariant Riemannian metric g G/N so that π : (G, g) → (G/N , g G/N ) becomes a Riemannian covering, that is, a covering map which is locally isometric.

Proposition 2.13 For i ∈ {1, 2}, let G i be a simply connected Lie group endowed with a leftinvariant Riemannian distance and let π
be a Riemannian covering as above. Then every bi-Lipschitz homeomorphism f :

where 'bi-Lipschitz' refers to the Riemannian distances induced by the respective Riemannian metrics.
Proof Let f : G 1 /N 1 → G 2 /N 2 be bi-Lipschitz. Since f is a homeomorphism and G 1 is simply connected, the composition f • π 1 : G 1 → G 2 /N 2 is a universal cover of G 2 /N 2 , as is the map π 2 : G 2 → G 2 /N 2 . It follows from the uniqueness theorem for universal covers, see for instance [10,Corollary 13.6] or [24, I, §11], that there exists a homeomorphism f : G 1 → G 2 with π 2 • f = f • π 1 . Since f is bi-Lipschitz and π 1 , π 2 are local isometries, the map f is uniformly locally bi-Lipschitz, as is its inverse. Finally, since G 1 and G 2 are geodesic, f is bi-Lipschitz as claimed.

Proposition 2.14 Each of the following pairs consists of quasi-isometrically homeomorphic groups that are not bi-Lipschitz equivalent:
Proof Once we know that R 2 × T 1 and N * 3 (R) are equivalent via a quasi-isometric but not a bi-Lipschitz homeomorphism, the same statements follow for S E(2) k and N * 3 (R) by Proposition 2.2, (2). Thus it suffices to prove Part (1) of Proposition 2.14.
In order to show that the groups N * 3 (R) and R 2 × T 1 are quasi-isometric via a homeomorphism, it is convenient to choose, as we may, coordinates (x, y, z) on N 3 (R) so that for all (x, y, z) and (x , y , z ), we have Without loss of generality we may assume that N * 3 (R) is the quotient of N 3 (R) by the cyclic group generated by the element Z = (0, 0, 1). The Lie group N 3 (R)/ Z is diffeomorphic to R 2 × T 1 . We see that Z 2 can be identified with a subgroup of the groups N 3 (R)/ Z and R 2 × T 1 , respectively, which in both cases acts co-compactly. Moreover, for these particular models, the identity map of R 2 × T 1 provides a quasi-isometric homeomorphism between N * 3 (R) and R 2 × T 1 .
Assume towards a contradiction that there exists a biLipschitz map f : R 2 ×T 1 → N * 3 (R). It follows from Proposition 2.13 that there would exists a bi-Lipschitz homeomorphism f : R 3 → N 3 (R). This is known to be false, for instance because R 3 has volume growth of order 3, whereas the volume of balls in N 3 (R) grows with order 4 at large. We have thus proven that N * 3 (R) is not bi-Lipschitz equivalent to R 2 × T 1 .

Proposition 2.15
Each of the following pairs consists of quasi-isometrically equivalent groups that are not equivalent via a quasi-isometric homeomorphism: Proof The groups appearing on the same line in Proposition 2.15 are topologically distinct and hence cannot be equivalent via a quasi-isometric homeomorphism. Indeed, denoting by " " equivalence via a diffeomorphism of manifolds, we have: 3) R 2 and R 2 × T 1 R 2 × S 1 (4) Aff + (R) R 2 and Aff + (R) × T 1 R 2 × S 1 (5) R 2 and N * 3 (R) R 2 × S 1 (6) R 2 and SE(2) k R 2 × S 1 (7) Aff + (R) R 2 and PSL(2) k R 2 × S 1 .
It remains to show that groups appearing on the same line are quasi-isometrically equivalent, even if they are not homeomorphic. First, the groups T 1 , T 2 , T 3 , SU(2), and SO(3) are trivially quasi-isometrically equivalent because they are compact.
Second, the groups R, R × T 1 , and R × T 2 are clearly quasi-isometrically equivalent. More generally, R × K is quasi-isometric to R × K for arbitrary compact Lie groups K and K , as one can see by arguing componentwise. For the same reason, R 2 and R 2 × T 1 are quasi-isometrically equivalent, and so are Aff + (R) and Aff + (R)×T 1 . Having established the quasi-isometric equivalence in the cases (1)-(4), the remaining cases follow by transitivity. Indeed, the information from Propositions 2.14, 2.2, and 2.10 can be used to deduce that the groups in (5), (6), and (7) are quasi-isometrically equivalent, once this has been established for the groups in (3) and (4).

Conclusion of the quasi-isometric classification
In Sect. 2 we have identified pairs of Riemannian Lie groups that are quasi-isometrically equivalent. In this section we show that all remaining pairs of at most three-dimensional connected Lie groups are quasi-isometrically distinct, thus establishing Theorem 1.2. The proof uses the following quasi-isometric invariants of connected Riemannian Lie groups: • degree of polynomial volume growth • polynomial volume growth (or equivalently by [17,19]: type R) • Gromov hyperbolicity [15], see also e.g. [29,Theorem 3.1.11] • covering dimension of asymptotic cones [6].
Besides these general quasi-isometry invariants, we also rely on quasi-isometric classification results for connected Riemannian Lie groups of a specific form: • for Gromov hyperbolic connected Riemannian Lie groups (which are proper metric spaces): topology of the boundary [15], see also e.g. [20,Proposition 2.20] • for simply connected Riemannian manifolds of negative or zero curvature: L p cohomology [16] • [30, Corollaire 1] and [36, Corollary 1.3] for R n A R with A ∈ Mat(n × n) having only eigenvalues with positive real parts (in our notation this applies to: J , D λ for 0 < λ ≤ 1, C λ for λ > 0) • [9, Theorem 1.3] for Sol(m, n), the solvable Lie groups R 2 R, where R acts by z·(x, y) = (e mz x, e −nz y), for m > n > 0 using coarse differentiation (in our notation: Sol(1, −λ) = D λ for −1 < λ < 0) Proof of Theorem 1. 2 We first discuss why the listed classes are quasi-isometrically distinct. The groups in classes (1)- (5) are the only groups of type R, as can be seen from an explicit description of the Bianchi classification of Lie algebras, as given for instance in [12,Chapter 7,§1.1]. The individual classes are divided according to the degree d ∈ {0, 1, 2, 3, 4} of polynomial volume growth.
The groups in classes (6) and (7 λ ) have exponential growth but are not Gromov hyperbolic: for the groups in class (6) this is easy to see since Aff + (R) × R can be endowed with a left-invariant Riemannian metric such that it contains an isometrically embedded copy of R 2 . The proof that D λ is not hyperbolic for λ < 0 is given below in Proposition 3.1.
We now show that (6) and (7 λ ) are distinct classes. The group Aff + (R) × R is not quasiisometrically equivalent to any D λ since the covering dimension of the asymptotic cone of D λ is 1 for every λ, while Aff + (R) × R has cone dimension 2 by [6,Theorem 1.1].
The groups in classes (8) This shows in particular that J cannot be quasi-isometric to any D λ , λ ∈ (0, 1], and D λ is quasi-isometric to D λ only if λ = λ . The previous discussion also covers the groups {C λ : λ > 0}, which are quasi-isometric to D 1 . Except for (7 λ ) and (11 λ ), which represent uncountably many different classes, all the groups listed on one line in the table in Theorem 1.2 are quasi-isometrically equivalent: this follows from Propositions 2.2, 2.11, 2.14, and 2.15.
We now discuss the proof of one result which has been used in the quasi-isometric classification. There are different proofs available for this fact. One can show for instance that the Dehn function of D λ , λ ∈ [−1, 0), is exponential (the argument for D −1 is outlined in [37]), and then use a result by Gromov [15] to deduce that D λ , λ ∈ [−1, 0) is not Gromov-hyperbolic since the Dehn function is not linear. Another possibility would be to consider the asymptotic cone of D λ , λ ∈ [−1, 0); see [4] and references therein. A proof for Proposition 3.1 is also contained in [9, §3.1], where it was observed that points in D λ , λ ∈ [−1, 0), which are not contained in the same hyperbolic plane can be joined by quasi-geodesics that do not lie close to each other. We recall the argument below. It is convenient to think of the hyperbolic plane H 2 not as the upper half plane {(u, v) : v > 0} with the metric given by but rather to apply a coordinate transform (x, z) = (u, log v). Then H 2 can be seen as R 2 equipped with the metric given by ds 2 = e −2z dx 2 + dz 2 . It turns out that the groups D λ , λ ∈ [−1, 0), are all foliated by isometrically embedded copies of H 2 . Perpendicular to these planes, there is another family of homothetically embedded 'upside down' versions of H 2 .

Proof of Proposition 3.1 following [9]
Recall that D λ is R 3 with the group law (x, y, z) * (x , y , z ) = (x + e z x , y + e λz y , z + z ).
Let g λ be the metric on D λ which makes the following left-invariant frame orthonormal: [Note that {E 1 , E 2 , E 3 } has structure constants as described in (1.2).] The associated length element is given by ds 2 = e −2z dx 2 + e −2λz dy 2 + dz 2 .
It follows that the planes {y = const} are isometrically embedded copies of H 2 , whereas the planes {x = const} are homothetically embedded copies of the reflected hyperbolic plane. Consider two points p 1 = (x 1 , y 1 , z 1 ) and p 2 = (x 2 , y 2 , z 2 ) in D λ with x 1 = x 2 and y 1 = y 2 . We will construct two quasi-geodesics γ a and γ b which connect p 1 and p 2 but do not lie close to each other. First, we let γ a,1 be the geodesic segment between p 1 and (x 2 , y 1 , z 2 ) inside the hyperbolic plane {y = y 1 }. Then we let γ a,2 be the geodesic segment in {x = x 2 } connecting the endpoint of γ a,1 to p 2 , and we denote by γ a the concatenation of γ a,1 and γ a,2 . The curve γ b is obtained in an analogous way, by first connecting p 1 to (x 1 , y 2 , z 2 ) by a geodesic segment in the plane {x = x 1 }, and then connecting the point (x 1 , y 2 , z 2 ) to p 2 by a geodesic in the hyperbolic plane {y = y 2 }. Observe that the map D λ → H 2 × H 2 , (x, y, z) → ((x, z), (y, z)) is a quasi-isometric embedding with constants depending only on the parameter λ if D λ is endowed with the distance induced by g λ and H 2 × H 2 is equipped with a product metric of d H 2 , where d H 2 is induced by a metric of constant sectional curvature equal to −1. It follows that both γ a and γ b are (L, C)-quasi-geodesics, for constants L = L(λ) ≥ 1 and C = C(λ) ≥ 0 independent of a and b. By applying this construction to a sequence of points p 1,n = (x 1,n , y 1,n , z) and p 2,n = (x 2,n , y 2,n , z), with z ∈ R, |x 1,n − x 2,n | → ∞ and |y 1,n − y 2,n | → ∞ as n → ∞, we see that there does not exist a constant δ > 0 such that for every n, the curve γ a connecting p 1,n to p 2,n is contained in the δ-neighborhood of γ b . This proves that (D λ , g λ ) is not Gromov hyperbolic.
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