Rough similarity of left-invariant Riemannian metrics on some Lie groups

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity. This allows us to reformulate in a common framework former results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner-M\"uller-Xie, on quasiisometries of these solvable groups.

1. Introduction 1.1.Main results.In this paper, we compare left-invariant Riemannian metrics on certain simply connected solvable Lie groups.The groups under study fall within two classes: • Heintze groups, that is, simply connected solvable groups with Lie algebra s such that n = [s, s] has codimension 1 in s and s splits as n R, where R acts on n via a derivation D whose eigenvalues have positive real parts.• Sol-type groups, that is, simply connected solvable groups with Lie algebra g such that n = [g, g] has codimension 1 in g and g splits as n R, where R acts on n via a derivation D whose eigenvalues have nonzero real parts, not all of the same sign, and such that Date: August 16, 2022.2010 Mathematics Subject Classification.20F69, 22E25, 53C23.Key words and phrases.Heintze groups, solvable groups, quasiisometry, rough similarity.All three authors were partially supported by the European Research Council (ERC Starting Grant 713998 GeoMeG 'Geometry of Metric Groups').Moreover, the great part of this work was done while the three authors were enjoying the stimulating environment of the University of Pisa.E.L.D. was also partially supported by the Academy of Finland (grant 288501 'Geometry of subRiemannian groups' and by grant 322898 'Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory') and by the Swiss National Science Foundation (grant 200021-204501 'Regularity of sub-Riemannian geodesics and applications').X.X. was partially supported by Simons Foundation grant #315130.
[n >0 , n <0 ] = 0, where n <0 (resp.n >0 ) is the sum of eigenspaces with negative (resp.positive) real part.Some relevant properties of these groups will be recalled in § 2; the main common feature of the groups we consider is to be simply connected, solvable, and have one-dimensional first cohomology, though the latter do not constitute a characterization.
In order to state our main result, recall that if φ : X → Y is assumed to be a quasiisometry between metric spaces X and Y , then for some c 0 there are positive constants λ − and λ + such that λ − d(x, x ) − c d(φ(x), φ(x )) and λ + d(x, x ) + c d(φ(x), φ(x )) for every x, x ∈ X. Say that the quasiisometry φ is a rough similarity if one can further take λ − = λ + and a rough isometry if one can take λ − = λ + = 1 in the inequalities above.
Theorem A. Let S be a Heintze group and let g 1 and g 2 be left-invariant Riemannian metrics on S with distance function d 1 and d 2 , respectively.Then the identity map Id : (S, d 1 ) → (S, d 2 ) is a rough similarity.
Theorem B. Let G be a Sol-type group and g 1 and g 2 be left-invariant Riemannian metrics on G with distance function d 1 and d 2 , respectively.Then the identity map Id : (G, d 1 ) → (G, d 2 ) is a rough similarity.
Theorems A and B imply the folowing statement at no cost: if ϕ is an automorphism of a Heintze or Sol-type group, then ϕ is a rough similarity with respect to any left-invariant Riemannian metric.(It is an elementary fact that the inner automorphisms of any group G equipped with a left-invariant distance d are rough isometries; however, group automorphisms are in general no more than quasiisometries assuming in addition that G is compactly generated and d is proper geodesic.) Using Theorems A and B we are able to reformulate certain results that appeared separately in the litterature.In the statement below, a Heintze group is of special type if it is a closed cocompact subgroup of a rank-one simple Lie group; Carnot type is a subclass of Heintze groups in which the nilradical is a Carnot group, and the derivation D is a Carnot derivation of this group.For the background on Carnot groups, see for example [LD17].The real shadow construction will be recalled along with precise definitions in § 2.3.
The substantial part of the following theorem is provided by the given references, while its formulation depends on the results above.
Theorem C. Let G belong to the following list: (1) The Lie group SOL [EFW13].
(2) Heintze group whose real shadow is of Carnot type with reducible first stratum [LDX16].
(3) Heintze group whose real shadow is not of Carnot type [CP17].
(4) (a) Heintze group whose real shadow is of Carnot type, which is different from the special-type subgroups in SO(n, 1) or SU(n, 1), and whose nilradical is nonrigid in the sense of Ottazzi-Warhurst [KMX21].(b) The Carnot-type Heintze group over the subgroup of unipotent triangular real n × n matrices, n 4 [KMX22].
Equip G with any left-invariant Riemannian metric with associated distance d.If φ : G → G is a quasiisometry, then φ is a rough isometry with respect to d.
Note that, in general, the notion of a rough isometry of a group does not make sense because it depends on the left-invariant distance one choses on the group.In view of Theorems A and B, the conclusion of Theorem C may also be stated in the following way: given any pair of left-invariant Riemannian distances d 1 and d 2 , every quasiisometry (G, d 1 ) → (G, d 2 ) is a rough similarity, whose similarity constant only depends on the pair (d 1 , d 2 ).
We point out that the rigidity property of quasiisometries expressed in Theorem C is weaker than the rigidity of quasiisometries (which means every self quasiisometry of a certain metric space is at a finite distance from an isometry).Every map at a finite distance from a isometry is a rough isometry.However, depending on the space there may exist rough isometries that are not at finite distance from any isometry, and this does actually happen for the left-invariant metrics on certain Heintze and Sol-type groups.
We also note the following: • Carrasco Piaggio has stated the conclusion in an equivalent form when G is as in (3) and additionally purely real [CP17].His result subsumes former ones, the first of which being by Xie and Shanmugalingam [SX12], the second one by Xie in [Xie14].• Case (2) subsumes former work by Xie in [Xie13].The groups of class (C) defined in [Pan89b,14.1]fall within this family (See Remark 5.2), and the early [Pan89b, Theorem 4] implies Theorem C for these: their quasiisometries are actually a bounded distance away from inner automorphisms.• Cases (2) and (4a) overlap, though none of them imply the other.The groups considered in [Pan89b, §14.3] belong to both classes.Case (4b) is not implied by (2) nor by (4a).• The statements in the references given are not uniform, so the degree of reformulation varies.
Bringing them together, the cases (2), (3) and (4) of Theorem C support the following conjecture: Conjecture D. Let S be a Heintze group, which is not among the special-type subgroups of SO(n, 1) or SU(n, 1) for any n 2. Equip S with any left-invariant Riemannian metric.Then every self-quasiisometry of S is a rough isometry.
We will discuss further the relations and differences of Theorem C and Conjecture D with quasi-isometric rigidity in the case of Heintze groups in §1.2.2.Especially, we will see there that Conjecture D would follow from conjectures already explicitely stated in [KMX21] and [Cor18].Keeping in mind that every homogeneous space of negative curvature is a Heintze group with a left-invariant metric, Conjecture D can be considered as a precise version of the feeling expressed in the four lines before §1 in [Pan89b].

Spaces of left-invariant metrics and comments on
Theorems A and B. The space of leftinvariant Riemannian metrics on a given Lie group has been widely studied by differential geometers; let us rather restrict our discussion to the results that put an emphasis on large-scale geometry rather than on Lie groups, for we believe that this comparison is more instructive.
For a finitely generated group Γ, Gromov introduced a metric space denoted by W M Γ whose points are word metrics and the distance is measured by the logarithm of λ, where (1/λ, λ) is the optimal pair of multiplicative quasiisometry constant between them [Gro93].The definition of this space itself is not straightforward, as one may consider several variants, especially one could compare metrics only through the identity map (as we do here), or through automorphisms, or even through arbitrary maps 1 .One may also include metrics that are not word metrics, especially geometric metrics, induced by the Riemannian metrics on universal covers when Γ is the fundamental group of a compact manifold.The resulting space is in some sort reminiscent of Teichmüller space, and actually contains it when Γ is a surface group.
Recently one of the variants of this space of left-invariant metrics was studied by Oregón-Reyes in the case of word hyperbolic groups [OR22, Theorem 1.3].Oregón-Reyes notes the analogy with Teichmüller spaces and identifies metrics that are roughly similar through the identity.
Theorem (Oregón-Reyes).Let Γ be a word-hyperbolic group.Consider the space D(Γ) of leftinvariant metrics on Γ that are quasiisometric to word metrics, modded out by the equivalence relation d ∼ d if d and d are roughly similar through the identity.Equip D(Γ) with the metric Then D(Γ) is unbounded.
All the Heintze groups being Gromov-hyperbolic, Oregón-Reyes result is in sharp constrast with ours, which suggests that Theorems A and B may be special to non-finitely generated groups.Whether they are special to connected Lie group is currently unknown to us and we ask specific questions in this direction at the end of this paper.1.2.2.Differences with other forms of rigidity.Some of the papers cited in Theorem C were dedicated to proving quasiisometric rigidity, and they are known for this, so that it may be useful to point out the differences of the conclusion of Theorem C with quasiisometric rigidity itself.Namely, the following is expected: QI Rigidity Conjecture.Let Γ be a finitely generated group.
(1) If Γ is quasiisometric to a Heintze group S, then S is of special type, and Γ is virtually a lattice in the rank-one simple Lie group containing S as a co-compact closed subgroup.(2) If Γ is quasiisometric to a Sol-type group G, then G is unimodular, and Γ is virtually a lattice in a Lie group G containing G as a co-compact closed subgroup.
A common significant ingredient between quasiisometric rigidity and Theorem C can be singled out in the case of Heintze groups that are not of special type.It is the following.
Pointed sphere Conjecture (Cornulier,[Cor18,19.104]).Let φ be a self quasiisometry of a Heintze group, not of special type.Then the extension of φ to the Gromov boundary of S fixes the unique boundary point that is fixed by all left-translations of S.
While conjectural in general, the following scheme of proof for Conjecture D should help the reader to understand our approach of some of the special cases of it in the present paper.
Proof of Conjecture D assuming the Pointed Sphere Conjecture and [KMX20, Conjecture 1.13].(See Figure 1).Let S = N R be a Heintze group as in the statement of Conjecture D. Then, 1 One should also decide if roughly isometric or roughly similar metrics are to be identified; however this is not a deep distinction.

Conjecture D ⇓
Theorem C (2), (3), (4) QI rigidity of special type Figure 1.Relation to QI rigidity.In this conjectural picture, the parts that are proved are framed within continuous lines.
• Either S is not of special type.In this case, the Gromov boundary of S can be identified with a one-point compactification of N , with the boundary extension of φ stabilizing N .By [KMX20, Conjecture 1.13], then, the boundary extension of φ to N equipped with a Carnot-Carathéodory metric should be bilipschitz, which, by [SX12], implies that φ is a rough isometry.for some n 2. The quasiisometries of S are at a bounded distance from isometries of a left-invariant symmetric Riemannian metric on S by [Pan89b], which implies by Theorem A that they are rough isometries of any left-invariant Riemannian metric, as mentionned in the paragraph below Theorem C.
The QI rigidity conjecture for Heintze groups, on the other hand, would follow from a combination of the QI rigidity for special-type groups, which were obtained in the 1980s and early 1990s (See Figure 1), together with the fact that no finitely generated group should be quasiisometric to a non-special Heintze group.We refer to [SX12, Proof of Corollary 1.3] for how the Pointed Sphere Conjecture implies the last statement.
Finally, an analogy coming from the world of finitely-generated groups may lead one to think of quasiisometries of groups as large-scale counterparts of homotopy equivalences between compact manifolds.Following this analogy, at least in nonpositive curvature, rough isometries are largescale counterparts to those homotopy equivalences that identify the marked length spectra (see e.g.[Fuj16]).Theorem C may then be considered analogous to the rigidity result that would consist in upgrading homotopy equivalence to marked length spectra isomorphism.Mostow's rigidity, which goes from homotopy equivalence to isometry, is strictly stronger, while length spectrum rigidity, which goes from the length spectrum to the isometry type, measures the difference.1.3.Organisation of the paper.Section 2 collects preliminary material, namely definitions and three lemmas from Gromov-hyperbolic geometry.Section 3 proves Theorem A and Section 4 proves Theorem B. Section 4 is the technical heart of the paper, and Theorem B is significantly harder to prove than Theorem A. In Section 5 we start by proving a special case of Cornulier's Pointed Sphere Conjecture, which is instrumental in the reformulation of the main theorem of [LDX16].Next, we prove the other cases of Theorem C. In Section 6 we point out that the conclusion of Theorem B does not hold in the Lamplighter group, and suggest a strengthening of the conclusion expressed by Theorems A and B which would be formulated in term of geometric actions that we did not reach in this paper.

Preliminary
2.1.Notation.If G, H, N, S are Lie groups then g, h, n, s are their Lie algebras.
2.2.Gromov-hyperbolic geometry.Let T be a tree, ξ ∈ ∂T a point in the ideal boundary, and x, y ∈ T .Then the intersection of the two rays xξ, yξ is also a ray: xξ ∩ yξ = zξ, where xξ, yξ branch off at z.The distance d(x, y) equals the distance from x to the branch point z plus the distance from y to the branch point z.A similar statement holds for all Gromov-hyperbolic spaces.
The following lemma follows easily from the thin triangle condition.We omit the proof.
Then there is a constant C depending only on δ, points x ∈ xξ, y ∈ yξ such that d(x , y ) ≤ C and the concatenation xx ∪ x y ∪ y y is a (1, C)-quasi-geodesic.Here xξ denotes any geodesic joining x and ξ; similarly for yξ, xx , x y , y y.In particular, |d(x, y) − (d(x, x ) + d(y, y ))| ≤ C. Furthermore, x , y can be chosen so that they lie on the same horosphere centered at ξ.
The next two lemmas are more involved, and will not be used before Section 4 where they serve as a preparation for the key step of Theorem B. The starting point is a well-known fact about simply connected Riemannian manifolds with sectional curvature bounded above by a negative constant: if p, q lie on the same horosphere then the length of every path joining p and q outside the horoball is at least exponential in d(p, q).For completeness, we provide a proof that is also true for Gromov-hyperbolic spaces.Lemma 2.2.Let X be a proper geodesic δ-hyperbolic space, ξ ∈ ∂X, S a horosphere centered at ξ, and B the horoball bounded by S. Then for every p, q ∈ S and every path c in X\B joining p and q, the length of c satisfies (c) ≥ 2 − C, where C is a constant depending only on δ.
Proof.Let γ be a geodesic between p and q and r be a "highest" point on γ, that is, for any Busemann function b centered at ξ, we have b(r) = min{b(x)|x ∈ γ}, see Figure 3.We claim B(r, d(p, q)/2 − 2C 2 ) ⊂ B for some constant C 2 depending only on δ.To see this, we first notice that d(r, p) ≥ d(p, q)/2 or d(r, q) ≥ d(p, q)/2.Without loss of generality we assume d(r, p) ≥ d(p, q)/2.Next we consider the path γ[p, r] ∪ rξ, where γ[p, r] denotes the segment of γ between p and r.Since r is a "highest" point on γ, it is clear that γ[p, r] ∪ rξ is a (1, C 1 ) quasi-geodesic from p to ξ for some constant C 1 depending only on δ.By the Morse Lemma2 , the Hausdorff distance between pξ and γ[p, r] ∪ rξ is bounded above by a constant C 2 depending only on δ.Hence d(r, x) ≤ C 2 for some x ∈ pξ.Let r ∈ pξ be the point at the same height as r, that is, b(r ) = b(r).Then d(x, r ) ≤ C 2 (comparing the Busemann function of x and r with respect to ξ) and so by the triangle inequality d(r, r ) The claim follows from this.
Let p ∈ γ between p and r such that d(r, p ) = d(p, q)/2 − 2C 2 , and q ∈ γ between r and q such that d(r, q ) = d(p, q)/2 − 2C 2 , see Figure 3. Then the path c = γ[p , p] ∪ c ∪ γ[q, q ] joins p and q and lies outside the ball B(r, d(p, q)/2 − 2C 2 ).By Proposition 1.6 on page 400 of [BH99], the length of c satisfies For any subset A ⊂ X as in the above lemma, let be the height change of points in A. Such a quantity can also be similarly defined for subsets of a Sol-type group since there is a notion of height in a Sol-type group.
Lemma 2.3.Let X be a proper geodesic δ-hyperbolic space, ξ ∈ ∂X, b a Busemann function based at ξ, S a horosphere centered at ξ, and B the horoball with boundary S. Let p, q ∈ S and c : [0, l] → X\B a path with c(0) = p, c(l) = q.Then, (1) The length of c satisfies (c) ≥ 2H(c) + 2 Here C is the constant from Lemma 2.2, especially it only depends on δ.
Proof.The lemma follows immediately from Lemma 2.2 when H(c) ≤ d(p, q).So we assume Now for each i = k by considering the height change we get (c| 2.3.Heintze groups.Given a derivation D on a Lie algebra n, we denote by n D R the Lie algebra obtained as a semidirect product n R where 1 ∈ R acts on n by the derivation D. Definition 2.4 (Heintze group).Let N be a nilpotent simply connected Lie group and let D be a derivation of n that has only eigenvalues with positive real parts and the smallest one has real part equal to one.A Heintze group is a simply connected solvable Lie group having Lie algebra n D R.
Heintze groups are Gromov-hyperbolic.Even better, they have at least one lef-invariant Riemannian metric of negative sectional curvature [Hei74], and this is a characterization among connected Lie groups.Definition 2.6 (Real shadow).Let D be a derivation of a real Lie algebra n.The derivation D may be decomposed into commuting components D = D ss,r +D ss,i +D n , where D ss,r is semisimple with a real spectrum, D ss,i is semisimple with purely imaginary spectrum, and D n is nilpotent, all being derivations ([LDG21, Corollary 2.6]).The real shadow of s = n D R is defined as Heintze groups with a real shadow of Carnot type may be characterized geometrically by the fact that the conformal gauge on their boundary at infinity minus the focal point contains a geodesic metric, indeed even a subRiemannian one.
2.4.Sol-type groups.We define below a class of solvable groups, the most prominent of which is the three-dimensional group SOL.
Definition 2.7 (Sol-type).Let N 1 , N 2 be a pair of simply connected nilpotent Lie groups.Let λ > 0. Let D 1 , D 2 be a pair of derivations of n 1 and n 2 , respectively, so that n 1 D R and n 2 D R are the Lie algebras of two Heintze groups S 1 and S 2 , i.e., the real parts of the eigenvalues of D 1 , D 2 are positive and they are normalized so that the smallest ones of each have . Sketch view of a Riemannian Sol-type group and two geodesics.Note that we do not assume that n 1 ⊥ n 2 .
real parts equal to one.The derivation D = D 1 ⊕ (−λD 2 ) acts on the Lie algebra n 1 × n 2 and the corresponding semi-direct product A Sol-type group is unimodular if and only if tr(D 1 ) = λ tr(D 2 ) (which does not depend on D 1 and D 2 chosen).
Similar to SOL, the group G is foliated by the left cosets of S i = N i R. Note that S 2 is a "upside down" Heintze group, while S 1 is right side up.See Figure 5.

Left-invariant Riemannian metrics on Heintze Groups
In this section we show that every two left-invariant Riemannian metrics on an Heintze group S = N R are roughly similar through the identity map, see Theorem A.
Lemma 3.1.Let S be a simply connected solvable Lie group and assume that N = [S, S] has codimension 1 in S. For every left-invariant Riemannian metric g on S, there exists a oneparameter subgroup c : S/N → S that is a geodesic such that ċ(0) ⊥ n and π • c is the identity on S/N , if π : S → S/N denotes the associated projection.
Proof.Let ∇ be the Levi-Civita connection of the left-invariant metric g on S. Then by Koszul's formula for the Levi-Civita connection (see e.g.(5.3) in [Mil76]), for every It follows that the one-parameter subgroup c generated by T with T ∈ n ⊥ and g(T, T ) = 1 is a geodesic of g, since ∇ ċ ċ = 0.
The key of the proof of Theorem A is Lemma 2.1 from the previous section and the fact that for every two one-parameter subgroups c 1 , c 2 of S not contained in N , every left coset of c 1 (R) is at bounded distance from a unique left coset of c 2 (R), see Lemma 3.2.
Let g 1 , g 2 be two left-invariant Riemannian metrics on a Heintze group S. Let c 1 and c 2 be the one-parameter subgroups associated to g 1 and g 2 respectively by Lemma 3.1.
In the case when c 1 and c 2 have the same image, the rest of the proof of Theorem A is quite simple.We shall treat this case in the next section; afterwards we consider the general case.
Proof of Theorem A when c 1 and c 2 have the same image.Observe that the height map S → R is 1-Lipschitz, where we equip S with d i and S/N with the Hausdorff distance Hausdist d i for i = 1, 2. From now on we decompose S topologically as a product N × R where c 1 (t) = (1 N , t) for all t ∈ R, and for all n ∈ N we denote c n the curve c n (t) = (n, t).By rescaling the metric g 1 and g 2 we may assume that c 1 = c 2 , and that they are unit speed geodesics for d 1 and d 2 .It follows from the normalization convention that for i = 1, 2, Hausdist d i on S/N is also the standard absolute value on R. A useful consequence is that if two subsets are at d i -Hausdorff distance bounded by H for some i, then so are their maximal heights also differ by H.
Let C be the constant from Lemma 2.1 for both d 1 and d 2 .We shall show that the identity map Id : (S, d 1 ) → (S, d 2 ) is a rough isometry.Let x = (n, t), x = (ñ, t) ∈ S. Our assumption implies that the curves c n and c ñ are unit speed minimizing geodesics with respect to both d 1 and d 2 .Because of Lemma 2.1, for each i = 1, 2 exists t i such that the path x to y.Since the identity map (S, d 1 ) → (S, d 2 ) is biLipschitz, the path β 2 is an (L, A)-quasi-geodesic in (S, d 1 ) from x to y, where L, A depend only on d 1 and d 2 .By the Morse Lemma, the Hausdorff distance between β 1 and β 2 in (S, d 1 ) is bounded above by a constant depending only on d 1 and d 2 .Comparing heights we see that |t 1 −t 2 | is bounded above by a constant depending only on d 1 and d 2 .Finally Lemma 2.1 implies that |d 1 (x, y) − d 2 (x, y)| is bounded above by a constant depending only on d 1 and d 2 .This finishes the proof of Theorem A when c 1 and c 2 have the same image.
3.1.The general case: c 1 and c 2 might have different images.In order to consider the general case in the proof of Theorem A, we need the following lemma.We shall abbreviate the image of R under a one-parameter subgroup c : R → S by c. Lemma 3.2.Let S be a Heintze group with derived subgroup N .Equip S with a left-invariant Riemannian metric g.For every two one-parameter subgroups c 1 , c 2 of S not contained in N , there is a positive number C (depending on c 1 , c 2 and g) such that for every s 1 ∈ S, there is a unique left coset s 2 c 2 of c 2 , with s 2 ∈ S, such that where Hausdist d denotes the Hausdorff distance with respect to the distance d on S determined by g.
Proof.Let g 2 be a left-invariant Riemannian metric on S such that c 2 is normal to N with respect to g 2 , and denote d 2 the associated Riemannian distance.For every s 1 ∈ S, the curve s 1 c 1 is an (L, C)-quasi-geodesic in (S, d 2 ) for some constants L, C depending only on g and g 2 .By the Morse Lemma, there is a complete geodesic γ in (S, d 2 ) such that Hausdist d 2 (s 1 c 1 , γ) ≤ C 1 for some constant C 1 depending only on g and g 2 .Since s 1 c 1 intersects all the horospheres centered at the focal point, so does γ (Indeed, h(s 1 c 1 ) and h(γ) are both intervals of R at bounded Hausdorff distance from each other, so if one of them is R then the other one as well).We see that the limit points of γ in ∂S are the focal point and some n ∈ N .On the other hand, there is a left coset s 2 c 2 with the same limit points.Since both γ and s 2 c 2 are geodesics in (S, d 2 ), their Hausdorff distance is bounded above by a constant H depending only on d 2 .Hence Hausdist The lemma follows as all the left-invariant Riemannian metrics are biLipschitz with respect to each other.Since two different cosets s 2 c 2 and s 2 c 2 have infinite Hausdorff distance, we have uniqueness.
Proof of Theorem A in the general case.For each i ∈ {1, 2}, let g i be a left-invariant Riemannian metric on S and d i the distance on S determined by g i .We need to show that the identity map (S, d 1 ) → (S, d 2 ) is a rough similarity.Let c i be a g i -geodesic section of π : S → S/N with c i (+∞) equal to the focal point for all i.The composition h • c i : R → R is the identity map.After rescaling the metric g i if necessary we may further assume that c i is a unit speed geodesic in (S, d i ).We shall show that the identity map (S, d 1 ) → (S, d 2 ) is a rough isometry.By symmetry it suffices to show that there is a constant C such that d 1 (x, y) ≤ d 2 (x, y) + C for every x, y ∈ S.
By Lemma 2.1, there are points where C depends only on d 2 .Since Id : (S, d 1 ) → (S, d 2 ) is L-biLipschitz for some L ≥ 1, we have By Lemma 3.2 there are left cosets α, β of c 1 such that Hausdist Considering the height function h, we take x and x to be points on α satisfying h(x) = h(x), h(x ) = h(x ), see Figure 6.Similarly let ỹ and ỹ be points on β satisfying h(ỹ) = h(y), h(ỹ ) = h(y ).We claim that we have for z ∈ {x, x , y, y } and the respective z.
Indeed, the d 1 distance from z to the appropriate left coset of c 1 is at most C, so that the height of the nearest-point projection of z on this left coset differs at most C from that of z.We have the bounds where we used the following arguments: In the first line, we used the triangle inequality.In the second line, we used (3.2) and (3.3).In the third line, we used that ) and similarly, d 1 (ỹ, ỹ ) = d 2 (y, y ).In the fourth line, we used (3.1).

Left-invariant Riemannian metrics on Sol-type groups
In this section we show that every two left-invariant Riemannian metrics with associated distances d 1 and d 2 on a Sol-type group G are roughly similar through the identity map, see Theorem B. Notation here is as in The general strategy is the same as for the Heintze groups: we first establish the statement for those pairs of Riemannian metrics for which n 1 × n 2 have the same orthogonal complement in g, then the general case.To establish the special case we need to find an estimate for the distance function, see Theorem 4.1.
4.1.Distances on Sol-type groups and the proof of Theorem B. In this subsection we state a result giving an estimate of distance in Sol-type groups and use it to prove Theorem B. The estimate itself will be established later.
Let g be a left-invariant Riemannian metric on G.By Lemma 3.1 we choose a geodesic section for G → R, and then assume without loss of generality that as a set, G = N 1 × N 2 × R, where R direction is perpendicular to both N 1 and N 2 with respect to g.As in the case of Heintze groups, this assumption implies that for every x ∈ N 1 , y ∈ N 2 , the curve γ x,y (t) = (x, y, t) (t ∈ R) is a minimizing constant speed geodesic.These will be called vertical geodesics.
We define several maps.The map h : G → R, h(x, y, t) = t, will be called the height function of G and t will be called the height of the point (x, y, t).The "projections" π i : G → S i are defined by π 1 (x, y, t) = (x, t), π 2 (x, y, t) = (y, t).We emphasize that the maps π i are not nearest point projections.However, they are Lie group homomorphisms and so are Lipschitz with respect to left-invariant Riemannian metrics, see Lemma 4.5.
By rescaling the metric g we may assume that the vertical geodesics are unit speed geodesics.Denote by d the distance on G determined by g.We identify S 1 with N 1 × {0} × R ⊂ G and S 2 with {0} × N 2 × R ⊂ G.For j = 1, 2, let g (j) be the Riemannian metric on S j induced by g and d (j) the associated distance on S j .
Define a "distance" ρ : It turns out that the distance d on G differs from ρ by a bounded constant: Theorem 4.1.Let G, d and ρ be as above.Then there exists a constant C > 0 such that |d(p, q) − ρ(p, q)| ≤ C for all p, q ∈ G.
As a consequence of the proof of Theorem 4.1 we have Corollary 4.2.Let G be a Sol-type group, g a left-invariant Riemannian metric and d the associated distance.Then there is a constant C > 0 with the following property.Denote by c the one-parameter subgroup of G that is perpendicular to N at e with respect to g.For every x, y ∈ G with h(x) ≤ h(y), there exist three left cosets β j (j = 1, 2, 3) of c with x ∈ β 1 , y ∈ β 3 and points x 1 ∈ β 1 , z 1 , z 2 ∈ β 2 and y 2 ∈ β 3 satisfying: (1) |d(x, y) Remark 4.3.Tom Ferragut has a result similar to Theorem 4.1, see Corollary 4.17 of [Fer20].These two results have overlap but do not imply each other.The result in [Fer20] is for horospherical products X Y of Gromov Busemann spaces X, Y .On one hand, horospherical products are more general than Sol-type groups.On the other hand, the factors X and Y in a horospherical product are "perpendicular" in some sense, while N 1 and N 2 in a Sol-type group is not assumed to be perpendicular to each other with respect to the metric g (without loss of generality the direction of the R factor is perpendicular to both N 1 and N 2 , but we do not assume that N 1 and N 2 are perpendicular).Proof of Theorem B assuming Theorem 4.1 in the case of equal vertical gedesics.One may decompose G as a product, G = N 1 × N 2 × R, in such a way that the direction of the R factor is perpendicular to both N 1 and N 2 with respect to g 1 and g 2 .After rescaling we may further assume that vertical geodesics have unit speed with respect to both g 1 and g 2 .We shall show that the identity map Id : i be the Riemannian metric on S j induced by g i , and let d i be the associated distance on S j .Also let ρ i be the "distance" (see (4.1)) on G corresponding to g i .By Theorem 4.1 there is a constant C > 0 such that |d i (p, q) − ρ i (p, q)| ≤ C for all p, q ∈ G. On the other hand, since the vertical geodesics have unit speed in (S, g i ), Theorem A implies |d 2 (π j (p), π j (q))| ≤ C for some constant C ≥ 0 and all p, q ∈ G.It follows from the definition of ρ i that |ρ 1 (p, q) − ρ 2 (p, q)| ≤ 2C and so |d 1 (p, q) − d 2 (p, q)| ≤ 2C + 2C for all p, q ∈ G.
For the general case, we need an analogue of Lemma 3.2 for Sol-type groups.
Lemma 4.4.Let c, c : R → G = (N 1 × N 2 ) R be one-parameter subgroups of G not contained in (N 1 × N 2 ) × {0} ⊂ G. Let g be any left-invariant Riemannian metric on G. Then there is a constant C depending only on g and c, c with the following property.For any left coset pc of c, there is a unique left coset qc of c such that Hausdist d (pc, qc) ≤ C.
Proof.Since c, c are not contained in (N 1 × N 2 ) × {0}, the compositions h • c and h • c are automorphisms of R. By composing c, c with suitable automorphisms of R we may assume h • c(t) = t and h • c(t) = t for t ∈ R. The one-parameter subgroups c and c now have the expressions: c(t) = (a 1 (t), a 2 (t), t), c(t) = (ã 1 (t), ã2 (t), t) for some functions a 1 , ã1 : R → N 1 , a 2 , ã2 : R → N 2 .
Let c be the one-parameter subgroup of G whose tangent vector at e is g-perpendicular to (N 1 × N 2 ) × {0} and h(c(t)) = t.After rescaling the metric g if necessary we may assume that c is a unit speed geodesic with respect to g.This normalization implies that if β is a left coset of c and p 1 , p 2 ∈ β, then d(p 1 , p 2 ) = |h(p 1 ) − h(p 2 )|.Similarly let c be the normalized one-parameter subgroup of G corresponding to g such that h(c(t)) = t.We observe that, if β is a left coset of c and β is a left coset of c, and p 1 , p 2 ∈ β, p1 , p2 ∈ α with h(p 1 ) = h(p 1 ), h(p 2 ) = h(p 2 ), then d(p 1 , p 2 ) = |h(p 1 ) − h(p 2 )| = d(p 1 , p2 ).We shall show that the identity map (G, d) → (G, d) is a rough isometry.

4.2.
Another expression for ρ.We next start the proof of Theorem 4.1 and Corollary 4.2.Up to an additive constant, ρ admits another expression which is more convenient for our purpose.We first fix some notation.Let G, g, d, d (j) and ρ be as in Subsection 4.1.We recall that g is a left-invariant Riemannian metric on G such that the R direction is perpendicular to both N 1 and N 2 with respect to g, the vertical geodesics γ x 1 ,x 2 (x 1 ∈ N 1 , x 2 ∈ N 2 ) are unit speed minimizing geodesics, and the minimal distance between the two "horizontal sets" Since S 1 , S 2 are Gromov-hyperbolic, there is some constant δ > 0 such that both (S 1 , d (1) ) and (S 2 , d (2) ) are δ-hyperbolic.
It is easy to see that the path γpq is a (1, H) quasigeodesic between p and q and that its length (γ pq ) ≤ d (1) (p, q) + H, with H 0 depending only on δ.Hence by stability of quasigeodesics in Gromov-hyperbolic spaces, for every length minimizing geodesic γ pq between p and q the Hausdorff distance between γpq and γ pq satisfies: where C depends only on δ.
Then ρ and ρ differ by at most a fixed constant.
Since ρ, ρ, 0 differ from each other by a fixed constant, the following gives an expression for these quantities up to a constant: for p = (x 1 , x 2 , t), q = (y 1 , y 2 , s): Proof of Corollary 4.2.We first consider the case when R is perpendicular to N with respect to g.In this case c = {0} × {0} × R and the left cosets of c are vertical geodesics γ x,y .Let p = (x 1 , x 2 , t) and q = (y 1 , y 2 , s) with t ≤ s.With the notation from above, the three left cosets are β 1 = γ x 1 ,x 2 , β 2 = γ x 1 ,y 2 , β 3 = γ y 1 ,y 2 .The points are p 1 = (x 1 , x 2 , t x 2 ,y 2 ), r 1 = (x 1 , y 2 , t x 2 ,y 2 ), r 2 = (x 1 , y 2 , t x 1 ,y 1 ), q 2 = (y 1 , y 2 , t x 1 ,y 1 ).Notice that the length 0 of α 0 satisfies | 0 − (d(p, p 1 ) + d(r 1 , r 2 ) + d(q 2 , q))| ≤ 2. Now the claim follows from Theorem 4.1 and the fact that 0 and ρ(p, q) differ by a bounded constant.Now let g be an arbitrary left-invariant Riemannian metric on G. Let c be a one-parameter subgroup of G that is perpendicular to N at e with respect to g.Then the above argument goes through with vertical lines replaced with left cosets of c, S 1 replaced with (N 1 × {0} × {0})c, and S 2 replaced with ({0} × N 2 × {0})c.
Proof.This follows from the fact that π j is a Lie group homomorphism.We shall only prove the case when j = 1 since the case for j = 2 is similar.It suffices to show that the operator norm of the tangential map ) is independent of the point p.
It is easy to check that the following diagram commutes for every x ∈ G: This leads to the commuting diagram of tangential maps: ) Deπ 1 Since the metrics g and g (1) are left invariant, the maps D x L x −1 and D π 1 (x) L π 1 (x −1 ) are linear isometries.Now the commuting diagram implies that D x π 1 : (T x G, g) → (T π 1 (x) S 1 , g (1) ) and D e π 1 : (T e G, g) → (T e S 1 , g (1) ) have the same operator norm.
Let p = (x 1 , x 2 , t) and q = (y 1 , y 2 , s) with t ≤ s.Let β : [0, l] → G be the arclength parametrization of a length minimizing geodesic from q to p. Let Lemma 4.6.There is a constant C 0 ≥ 0 independent of the points p, q such that max{D Proof of Theorem 4.1 assuming Lemma 4.6.We use the fact that the minimal distance between Since 0 is the length of a curve between q and p, we have d(p, q) ≤ 0 .We shall show that the reverse inequality holds up to an additive constant by using the expression for ρ.Here we only write down the details for the case t x 2 ,y 2 ≤ min{t, s}, t x 1 ,y 1 ≥ max{t, s} as the other cases are similar.First assume β reaches height h + before it reaches height h − .From q the curve β first reaches the height h + , so this subcurve has length at least h + − h(q).Then β goes down to the height h − , so the length of this portion of β is at . Geodesic segment β between p and q in coordinate view.least h + − h − .Finally β goes up and reaches the height h(p), so the length of this portion of β is at least h(p) − h − .Hence the length of β is at least If β reaches height h − before it reaches height h + , then a similar argument shows that its length is greater than the quantity above.
Proof of Lemma 4.6.We will only consider the case D + ≥ D − and show that D + ≤ C 0 .The case D − ≥ D + can be similarly handled by considering π 2 instead of π 1 .We may assume that with C the constant from (4.2), otherwise we are done.
. Metric view of the four vertical geodesics involved.
We consider three cases depending on the value of h + − D + .
Case I: h + − D + > h(q).We will divide the curve β into several subcurves.Let There may be more than one such l 0 ; we just pick one.)(1) Subcurve . By considering the height change we see that (β 1 ) ≥ 1 .Remember that h(p) ≤ h(q) by assumption.Write Indeed, by the triangle inequality and the fact that the d (1) -geodesic γ ∩ between (x 1 , h + − D + ) and (y 1 , h + −D + ) lies in a C-neighborhood of the path γ defined as a concatenation of vertical geodesics and length-minimizing segment in the horosphere of height t x 1 ,y 1 between the same points (See Figure 13), we have that by the triangle inequality one of the following holds: Since β is a minimizing geodesic between q and p and 0 is the length of a path between p and q we have (β) ≤ 0 .On the other hand, Condition (I.1) or (I.2) or (I.3) holds.
Let β1 be the complement of β1 in β 1 .Since the height change of β 1 is at least 1 and the height change of β1 is at most 4d (1) (p , q ), we see the length of β1 satisfies ( β1 ) ≥ 1 − 4d (1) (p , q ).Together with the estimate of the length of β1 from the above paragraph we get (β where again for the last inequality we used d (1) (p , q ) ≥ D + /5.Now it is clear that d (1) (p , q ) and so D + is bounded above by a constant depending only on L, C, and δ.This finishes the proof of the Claim in Case I, Condition (Condition I.1).Now assume (Condition I.2) holds.This condition implies d (1) lies below the height h + , an argument similar to the case of (Condition I.1) finishes the proof.
As before d (1) ((x 1 , h + − D + ), (y 1 , h + − D + )) ≥ 39 10 D + and so by the triangle inequality one of the following holds: First assume (Condition II.1) holds.This implies d (1) lies below the height h + and we can repeat the argument in Case I to finish the proof.
Finally we assume (Condition II.2) holds.In this case the curve α 3 := γ x 1 ,x 2 | [h(p),h + −D + ] ∪ β 3 lies below the height h + − D + and we can repeat the argument in Case I to finish the proof.

5.1.
A special case of the pointed sphere conjecture.We shall refer here to the pointed sphere conjecture of Cornulier recorded in [Cor18,Conjecture 19.104].
By first stratum of a Carnot algebra with Carnot derivation D, we mean the eigenspace ker(D − 1), which by assumption Lie generates the Carnot algebra, see [LD17].The higher strata are the subspaces ker(D − i) for i 2; the Lie algebra is a direct sum of its strata.(One also encounters the term layer in the literature.) Let N be a Carnot group with Lie algebra n and first stratum V 1 .We say that N (or equivalently n) has reducible first stratum if there is a nontrivial subspace W of V 1 such that for every strata-preserving automorphism φ of n one has φ(W ) = W .Such a notion has been studied in [Xie13], however, the reader should not mistake it with the notion of reducibility, from the same paper.
Proposition 5.1.Let S = N R be a Heintze group of Carnot type.Assume that N has reducible first stratum.Then, the pointed sphere conjecture holds for S. Namely, every quasisymmetric self-homeomorphism of ∂S fixes the focal point in ∂S.
Proof.The argument that we shall follow is similar to the one in [LDX16] and it is based on [Xie13], such a principle goes back to [Pan89a, before Corollaire 6.9].Namely, we shall prove that the focal point in ∂S is fixed by proving that a special foliation in ∂S is preserved (See Figure 15).
Let ω be the focal point in ∂S, so that ∂S = N ∪ {ω}.Let F : ∂S → ∂S be a quasi-symmetric homeomorphism.We need to prove that F (ω) = ω.Let us assume that this is not the case.
Since N is assumed to have reducible first stratum V 1 , then there is a nontrivial subspace W of V 1 that is fixed by (the differential of) every strata-preserving automorphism of N .Consequently, the nontrivial group G generated by exp(W ) is preserved by every strata-preserving automorphism of N .The cosets of G induce a singular foliation on ∂S.In particular, when we restrict to the sets U 1 := N \ F −1 (ω) and U 2 := N \ F (ω), we have that the leaves on U 1 (resp.on U 2 ) are exactly the left cosets xG, as x ∈ N , except for a leave, which is . At this point we stress that in ∂S while F −1 (ω) and F (ω) are in the closure of just one of these leaves, the point ω is in the closure of every leave.
Let us restrict to the map F : Proof.Let g 0 be a left-invariant Riemannian metric on S = N R such that the N direction is perpendicular to R, let d 0 be the associated distance.Then the ideal boundary ∂S can be identified with N ∪{ω}, and the Carnot metric on N is a parabolic visual metric with respect to ω.Now let g be an arbitrary left-invariant Riemannian metric on S with d the associated distance, and let φ be a self quasiisometry of (S, d).Then φ is also a self quasiisometry of (S, d 0 ).By Proposition 5.1, ∂φ : ∂S → ∂S fixes the focal point of S and so induces a self quasisymmetric map ∂ * φ of N (with the Carnot metric).Then ∂ * φ is a biLipschitz homeomorphism of N [LDX16, Theorem 1.2].However, a self quasiisometry of a Gromov-hyperbolic space is a rough isometry if and only if the induced boundary map is biLipschitz.This follows from the results of Bonk-Schramm of n is equal to RD itself.Since the centralizer of RD Lie generates the strata-preserving automorphisms of n, and since the first stratum V 1 of a nilpotent Lie algebra has dimension greater than or equal 2, the first stratum of a Lie algebra of class (C) has a vector w 1 generating a proper subspace that is invariant under the strata-preserving automorphisms (in fact, any nonzero vector in the first stratum is good for this).It follows that the groups in Pansu's class (C) have reducible first stratum.Pansu proved that among Carnot groups N of class 2 with dim V 1 even, greater or equal to 10 and 3 Theorem 5.4 (After Ferragut).Let G be a non-unimodular Sol-type group.Equip G with any left-invariant Riemannian distance.Then, every self-quasiisometry of G is a rough isometry.
Proof of Theorems 5.3 and 5.4.Start assuming that G is a non-unimodular Sol-type group.Equip G with a horospherical product Riemannian metric g 0 , that is, a metric for which n 1 ⊥ n 2 .Decompose G = N 1 × N 2 × R where the direction of the R factor is g 0 -perpendicular to N 1 × N 2 .By [Fer22, Theorem 10.3.2],every quasiisometry Φ of G is a bounded distance away from (Ψ 1 , Ψ 2 , Id R ), where Ψ i is bilipschitz with respect to the D i -parabolic metric on N i .Using again [SX12] (Ψ 1 , Id R ) is a rough isometry of S 1 while (Ψ 2 , Id R ) is a rough isometry of S 2 .And then we conclude by Theorem 4.1 that Φ is a rough isometry of (G, g 0 ).Then by Theorem B, Φ is a rough isometry of G with respect to any left-invariant Riemannian distance.Now, let G be the three-dimensional Lie group SOL equipped with its standard metric written in coordinates (n 1 , n 2 , t) as ds 2 = e −2t dn 2 1 + e 2t dn 2 2 + dt 2 , and Φ is a quasiisometry of G, it follows from [EFW13] that up to possibly composing Φ with an isometry, Φ is at bounded distance from a product map of the form above.The end of the argument is the same as before.5.4.Restatement of Carrasco Piaggio's theorem.Theorem 5.5 (After Carrasco Piaggio).Let S be a Heintze group.Assume that the real shadow of S is not of Carnot type.Equip S with a left-invariant Riemannian metric.Then, every self quasiisometry of S is a rough isometry.
Proof.Let g 0 be the left-invariant Riemannian metric on S which is simultaneoulsy isometric to a left-invariant metric g0 on the real shadow S 0 [Ale75]; denote by ρ : S → S 0 any such isometry.By the published version of [CP17, Corollary 1.8], every self quasiisometry of S 0 is a rough isometry.Let φ be a self quasiisometry of S. Then ρφρ −1 is a self quasiisometry of S 0 , hence a rough isometry of S 0 with respect to g0 .It follows that φ is a rough isometry of g 0 , and then of any left-invariant metric by Theorem A. 5.5.Restatement of Kleiner-Müller-Xie's theorems.Theorem 5.6 (After Kleiner, Müller and Xie).Let S be a Heintze group whose real shadow is of Carnot type.Assume that the nilradical N = [S, S] is nonrigid in the sense of Ottazzi-Warhurst, and that N is not R d or a Heisenberg group.Equip S with a left-invariant Riemannian metric.Then, every self quasiisometry of S is a rough isometry.Z/mZ Z for m 2 share their asymptotic cones (namely, horospherical products of two R-trees equipped with a preferred horofunction, see [Cor08, Section 9]) with that of the group SOL, and their large-scale geometries are in many respect comparable.However we will prove below that they fail to have their word metrics roughly similar through the identity map.
Consider the following infinite presentation of the group L m : a, t | a m , [t i at −i , t j at −j ], i, j ∈ Z and the two finite generating sets • The wreath product generating set S w = {a, t} • The automaton generating set S a = {t, ta}.
We denote by d w and d a the word distances with respect to S w ∪ S −1 w and S a ∪ S −1 a respectively.In the following, by "color" we mean an element of Z/mZ.An element of L m is encoded by a lighting function Z → Z/mZ together with the position of a cursor, with the following multiplication law: Multiplying by t on the right amounts to moving the cursor to the right, and multiplying by a on the right amounts to shifting color at the position of the cursor.Let n be a positive integer (think of it large enough).Let us first consider the element g ∈ L m for which the cursor is located at 0 ∈ Z and the bulb at position n is lit with the color 1 ∈ Z/mZ (all the others being not lit).Note that g = t n at −n .Since in both generating sets, the cursor moves at most by one unit at each multiplication by a generator, the distance from 1 to g in both word metrics must be at least 2n.In fact, one computes that d w (1, t n at −n ) = 2n + 1 while t n at −n = (ta) n a(ta) −n = (ta) n t −1 (ta)(ta) −n = (ta) n t −1 (ta) −(n−1) hence d a (1, t n at −n ) = 2n.It follows that if d a and d w were to be roughly similar through the identity, they should differ by a constant.However, d w (1, (ta) n ) = 2n while d a (1, (ta) n ) = n as may be proved by counting the occurrences of t and the number of bulbs lit in the final configuration.
Remark 6.1.The wreath product metric is easier to undersand, and there are explicit formulae for the word length for families of words in normal form.Taback and Cleary have investigated the geometry of the automata metric and the result above could be deduced from their paper [CT05].
6.2.In search of a coarse notion.One of the main limitations of our present work is that the property that we identify, namely having all the left-invariant Riemannian metrics roughly similar, is not a coarse property.Indeed, Riemannian metrics play no special role among proper geodesic metrics as far as large-scale geometry is concerned.
The search for a coarse notion leads to the following considerations.Let G be a locally compact, compactly generated group.Denote by Geom(G) the collection of geometric actions, that is, pairs (X, α) where X is proper geodesic and α : G → Isom(X) is continuous, proper and cocompact.For every pair {(X, α), (Y, β)} in Geom(G), and for every pair of points o X ∈ X, o Y ∈ Y , the map G. α o X → G. β o Y determined by the identity map of G is a quasiisometry X → Y .We call this map (to be considered only up to bounded distance) the G-orbital map.If H < G is closed and co-compact, then H is still compactly generated locally compact [CdlH16, 2.C.8(3)], and there is a natural map Geom(G) → Geom(H) obtained by (X, α) → (X, α |H ).If K < G is a compact normal subgroup, and π : G → G/K is the associated epimorphism, then there is a natural map Geom(G/K) → Geom(G) obtained by (X, α) → (X, α • π).We essentially proved the following.Proposition 6.1.Let H be a compactly generated locally compact group.Assume that for every pair {(X, α), (Y, β)} in Geom(H), the orbital map X → Y is a rough similarity.Then (1) If K is a compact normal subgroup of H, then for every pair {(X, α), (Y, β)} in Geom(H/K), the orbital map X → Y is a rough similarity.(2) If G receives an injective homomorphism with closed and co-compact image from H, then for every pair {(X, α), (Y, β)} in Geom(G), the orbital map X → Y is a rough similarity.
Note that two-ended groups have the property in the proposition.Also, if Γ = H is a finitely generated group which sits as a uniform lattice in a locally compact group G, the property expressed by Proposition 6.1 transfers from Γ to G, but not from G to Γ. 6.3.Final questions.An affirmative answer to the next question would provide a robust generalization of the main results of the present paper.Question 6.2.Let G be a completely3 solvable Lie group with H 1 (G, R) = R.Does it hold that for every pair {(X, α), (Y, β)} in Geom(G), the orbital map X → Y is a rough similarity?Question 6.3.Same question as above, where G = (Z m × Z m ) Z is the locally compact group that contains L m as a lattice.

Figure 2 .
Figure 2. Lemma 2.1 in a tree and in the hyperbolic plane.
Definition 2.5 (Carnot-type Heintze group).A Heintze group S is of Carnot type if ker(D − 1) Lie generates n; this does not depend on the derivation D such that s n D R. The rank-one type Heintze groups defined in the Introduction are of Carnot type.A Heintze group has a distinguished family of horospheres, disregarding the choice of a particular left-invariant Riemannian metric.Those are left cosets of the derived subgroup N .By focal point of a Heintze group we mean the limit point of the subgroup N = [S, S] in the Gromov boundary.When S is naturally acting on its Gromov boundary, this point is the only one fixed by S.
2.5.Height.Definition 2.8.Let S = N D R be a Heintze group as in Definition 2.4.The projection h : S → R is called the height function of S. Definition 2.9.Let G = N D R be a Sol-type group as in Definition 2.7.The projection h : S → R is called the height function of G.

Figure 6 .Figure 7 .
Figure 6.Main objects in the proof of Theorem A in the general case, in the hyperbolic disk model with a focal point ω.From the point of view of d 2 , d 1geodesics appear in the form of hypercircles.

Figure 8 .
Figure 8.The left cosets of c in Corollary 4.2.

Figure 15 .
Figure 15.Invariant foliation on the boundary at infinity.
U 1 and U 2 are open set of the Carnot group N , by Pansu's differentiability theorem [Pan89b], the map F is Pansu differentiable at almost every point and its Pansu differential is a strata-preserving automorphism, which therefore preserves the proper subgroup G.By the argument in [Xie13, Proposition 3.4] we have that F preserves the leaves of the foliation that we are considering.Since we had a topological characterization of the point ω (and since the map F is continuous), then we get to a contradiction unless F fixes ω. 5.2.Restatement of Le Donne-Xie's theorem.Theorem 5.1 (After Le Donne-Xie).Let N be a Carnot group with reducible first stratum.Let S = N R be the Carnot-type Heintze group associated to N , and equip S with any left-invariant Riemannian distance.Then, every self-quasiisometry of S is a rough isometry.
[BS11] (Theorems 7.4 and 8.2).For a direct proof in the case of parabolic visual metric see [SX12, Lemma 5.1].Hence φ : (S, d 0 ) → (S, d 0 ) is a rough isometry.By Theorem A the identity map Id : (S, d 0 ) → (S, d) is a rough isometry.It follows that φ : (S, d) → (S, d) is also a rough isometry.Remark 5.2.Pansu defined the Carnot-type groups of class (C) in [Pan89b, 14.1].At the Lie algebra level, the definition reads as follow: the Carnot-type group N D R (where N is different from R) is of Pansu's class (C) if the centralizer of RD in the Lie algebra of derivations the property of being of class (C) is generic in the sense of algebraic geometry[Pan89b].This implies that having reducible first stratum is also a generic property among these groups.As mentionned in the Introduction, Theorem 5.1 for the groups in Pansu's class (C) is due to Pansu.5.3.Restatement of Eskin-Fisher-Whyte's and Ferragut's theorems.Theorem 5.3 (After Eskin-Fisher-Whyte).Let G be the Lie group SOL.Equip G with any left-invariant Riemannian distance.Then, every self-quasiisometry of G is a rough isometry.

Proof.
The pointed sphere conjecture holds for these groups by [KMX21, Theorem 1.2] and global quasisymmetric homeomorphisms of the boundary minus the focal point are bilipschitz by [KMX21, Theorem 3.1].The mechanism of proof is then exactly the same as for Theorem 5.1.Theorem 5.7 (After Kleiner, Müller and Xie).Let S be a Heintze group whose real shadow is of Carnot type.Assume that the nilradical N = [S, S] is the group of unipotent triangular real n × n matrices, n 4. Equip S with a left-invariant Riemannian metric.Then, every self quasiisometry of S is a rough isometry.Proof.Global quasisymmetric homeomorphisms of the boundary minus the focal point are bilipschitz by [KMX22, Theorem 1.3].By [KMX22, Corollary 3.2], there is an automorphsim τ of N , such that possibly after composing with τ , any local quasiconformal homeomorphism of the boundary locally preserve a coset foliation.The pointed sphere conjecture for S can be deduced in the same way as we did in §5.1.6. Limitations of the present work and questions left open 6.1.Failure of the analogous property for Lamplighter groups.The groups L m =