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Rigidity of fiber-preserving quasisymmetric maps

We show that fiber-preserving quasisymmetric maps are biLipschitz. As an application, we show that quasisymmetric maps on Carnot groups with reducible first stratum are biLipschitz.


Introduction
In this paper we study the rigidity property of quasisymmetric maps that preserve a foliation. We show that, under quite general conditions, such quasisymmetric maps are biLipschitz. We then give an application to quasisymmetric maps between Carnot groups.
Quasisymmetric maps that preserve a foliation arise when one studies the rigidity property of quasiisometries between negatively curved solvable Lie groups, see [Pi], [SX], [X1], [X4]. It is well known that a negatively curved space has an ideal boundary, and quasiisometries between negatively curved spaces correspond to quasisymmetric maps between the ideal boundaries. Rigidity properties of the quasiisometries correspond to the rigidity properties of the quasisymmetric maps. The ideal boundary of a negatively curved solvable Lie group is a nilpotent Lie group equipped with a homogeneous distance. Recent results suggest that very often the quasisymmetric maps on such nilpotent Lie groups must be biLipschitz [P], [CO], [CR], [SX], [X1], [X2], [X4], [X5].
There are usually two steps in the proof of the above mentioned rigidity property of the quasisymmetric maps. The first step is to show that the quasisymmetric map preserves a certain foliation. The second step is to show that a quasisymmetric map must be biLipschitz if it preserves a foliation. The main purpose of this paper is to take a closer look at the second step in the context of quasisymmetric maps between general metric spaces. We hope that this will be useful in the eventual (hopefully) complete solution of the rigidity of quasiisometries between negatively curved solvable Lie groups.
One often has to deal with quasimetrics instead of metrics while studying the ideal boundary of negatively curved spaces. For this reason we state our main result for quasimetric spaces. Recall that a function d : X × X → [0, ∞) is a quasimetric on a set X if (1) d is symmetric, that is, d(x 1 , x 2 ) = d(x 2 , x 1 ) for all x 1 , x 2 ∈ X; (2) d(x 1 , x 2 ) = 0 if and only if x 1 = x 2 ; (3) there exists some constant M ≥ 1 such that d(x 1 , x 3 ) ≤ M · (d(x 1 , x 2 ) + d(x 2 , x 3 )) for all x 1 , x 2 , x 3 ∈ X. We call M a quasimetric constant of d. A subset A of a quasimetric space X is called closed if for any x ∈ X and any sequence {a j } ⊂ A, the condition d(a j , x) → 0 implies x ∈ A.
To state our main result, we introduce the notion of "fibered quasimetric space". To do that we first specify the concept of parallelism for sets. We say that two closed subsets U and V of a quasimetric space X are parallel if there is a constant a > 0 such that d(u, V ) = d(v, U ) = a for every u ∈ U and every v ∈ V . Recall that d(u, V ) := inf{d (u, v) : v ∈ V }. It is easy to check that in this case we have d(U, V ) = HD(U, V ) = a, where and HD(U, V ) denotes the Hausdorff distance between U and V : The following definition provides the setting for our main theorem.
Definition 1.1 (Fibered quasimetric space). Let α > 0, L ≥ 1. We say that a quasimetric space X is an (α, L)-fibered quasimetric space if X admits a cover U by closed pairwise disjoint subsets, called fibers, with the following properties: Fibers are snow-flake equivalent to unbounded geodesic spaces: (1.2) for each U ∈ U , there exists an unbounded geodesic space ( U , d) such that U is L-biLipschitz to ( U , d α ); Parallel fibers are not isolated: (1.3) for any U ∈ U , there exists a sequence U i ∈ U such that U i and U are distinct, parallel, and HD(U i , U ) → 0 as i → ∞; Fibers have positive distance: Non-parallel fibers diverge: Given K ≥ 1 and C > 0, we say that a bijection F : X → Y between two quasimetric spaces is a (K, C)-quasi-similarity if Clearly a map is a quasi-similarity if and only if it is biLipschitz. The point here is that often there is control on K but not on C. In this case, the notion of quasi-similarity provides more information about the distortion.
The following is our main result and states that any fiber-preserving quasisymmetric map between fibered quasimetric spaces is a quasi-similarity (and hence biLipschitz). See Section 2.1 for the definition of quasisymmetric map.
Theorem 1.1. Let X, Y be (α, L)-fibered quasimetric spaces for some α > 0 and L ≥ 1. Suppose F : X → Y is an η-quasisymmetric map that sends fibers of X homeomorphically onto fibers of Y . Then F is a (K, C)-quasi-similarity, where K depends only on η, α, L and the quasimetric constants of X, Y .
We remark that for the validity of Theorem 1.1 the condition d(u, V ) = d(v, U ) = a for parallel fibers can not be replaced by the weaker condition HD(U, V ) < ∞. See the end of Section 3 for an example.
Theorem 1.1 in particular implies that all quasiconformal maps of the subRiemannian Heisenberg groups that send vertical lines to vertical lines must be biLipschitz, see Proposition 4.5. See [T] and [BHT] for the construction of such maps. We remark that there exist biLipschitz maps of Heisenberg groups that send vertical lines to curves that are not vertical lines, see [X3].
Our main application of Theorem 1.1 is to quasisymmetric maps on Carnot groups with reducible first stratum.
Let G be a Carnot group. See Section 2.2 for a brief introduction to Carnot groups. Let V 1 ⊕ · · · ⊕ V r be the stratification of the Lie algebra g of G. For each t > 0, the standard . . , r. Let Aut * (g) be the group of strata-preserving automorphisms of g. We say V 1 is reducible (or the first stratum of g is reducible) if there is a non-trivial proper linear subspace W 1 ⊂ V 1 such that A(W 1 ) = W 1 for every A ∈ Aut * (g).
The main consequence of Theorem 1.1 is the following theorem. Recall that a Carnot group is a metric space equipped with a particular Carnot-Carathéodory distance, see Section 2.2.
Theorem 1.2. Let G be a Carnot group with reducible first stratum. Then every quasisymmetric map F : G → G is a quasi-similarity, quantitatively.
Theorem 1.2 is quantitatively in the sense that for any Carnot group G and for any η ∈ Homeo([0, ∞)) there exists a constant K depending only on G and η such that every η-quasisymmetric map F : G → G is a (K, C)-quasi-similarity for some C.
We remark that in [CO], Cowling and Ottazzi proved that all smooth quasiconformal maps on rigid Carnot groups are affine maps (an affine map is the composition of a left translation and a strata-preserving automorphism) and so in particular are biLipschitz. On the other hand, it is shown in [X5] that if N is a non-rigid Carnot group other than the Euclidean groups and some very particular type of product groups, then every quasisymmetric map of N is biLipschitz. We note that Carnot groups with reducible first stratum include both rigid and non-rigid groups. In Theorem 1.2 there is no assumption on the regularity of the quasisymmetric map.
In Section 2 we recall some basic definitions and facts that are used later. In Section 3 we prove Theorem 1.1. In Section 4 we prove Theorem 1.2 and also give two other applications of Theorem 1.1: quasiconformal maps of Heisenberg groups that send vertical lines to vertical lines, and a new proof of a theorem of Dymarz [D] on quasisymmetric maps of ideal boundary of certain amenable hyperbolic locally compact groups.

Preliminaries
In this section we collect definitions and results needed later. We first recall the definitions of quasiconformal and quasisymmetric maps (Section 2.1). Then we review the basic definitions related to Carnot groups in Section 2.2, the BCH formula in Section 2.3 and Pansu Differentiability Theorem in Section 2.4.

Quasiconformal and quasisymmetric maps
A map F : X → Y between two quasimetric spaces is continuous if for any x ∈ X and any sequence {x j } ⊂ X, the condition lim j→∞ d(x, x j ) = 0 implies lim j→∞ d(F (x), F (x j )) = 0.
Let F : X → Y be a bijection between two quasimetric spaces. For x ∈ X and t > 0, define The map F is called λ-quasiconformal if both F and F −1 are continuous and A bijection F : X → Y between two quasimetric spaces is η-quasisymmetric if both F and F −1 are continuous and for all distinct triples x, y, z ∈ X, we have [V,Theorem 6.3]. A bijection between quasimetric spaces is said to be quasisymmetric if it is η-quasisymmetric for some η.
We remark that quasisymmetric maps between general quasimetric spaces are quasiconformal. In the case of Carnot groups, and more generally Loewner spaces, a quasiconformal homeomorphism is locally quasisymmetric, see [HK,Theorem 4.7].

Stratified Lie algebras and Carnot groups
A stratified Lie algebra is a finite-dimensional Lie algebra g over R together with a direct sum The integer s is called step or degree of nilpotency of g. Every stratified Lie algebra g = V 1 ⊕ V 2 ⊕ · · · ⊕ V s admits a one-parameter family of automorphisms λ t : We call such isomorphisms strata preserving.
Let G be a connected, simply connected Lie group whose Lie algebra is stratified as We fix a left-invariant inner product on HG. An absolutely continuous curve γ in G whose velocity vector γ ′ (t) is contained in H γ(t) G for almost every t is called a horizontal curve. Any horizontal curve has an associated length defined using the left-invariant inner product, i.e., by integrating the norm of its tangent vector. Since V 1 generates the whole Lie algebra, any two points of G can be connected by horizontal curves. Let p, q ∈ G, the Carnot-Carathéodory metric between p and q is denoted by d c (p, q) and is defined as the infimum of length of horizontal curves that join p and q. We call Carnot group the data of G, its stratification, the inner product on its first stratum, and consequently the distance d c .
Since the inner product on HG is left invariant, the Carnot metric on G is left invariant as well. Different choices of inner product on HG result in Carnot metrics that are biLipschitz equivalent. The Hausdorff dimension of G with respect to a Carnot metric is given by s i=1 i · dim(V i ). Recall that, for a simply connected nilpotent Lie group G with Lie algebra g, the exponential map exp : g → G is a diffeomorphism. Under this identification the Lesbegue measure on g is a Haar measure on G. Furthermore, the exponential map induces a oneto-one correspondence between Lie subalgebras of g and connected Lie subgroups of G.
Let G be a Carnot group with Lie algebra g = V 1 ⊕ · · · ⊕ V s . Since λ t : g → g (t > 0) is a Lie algebra automorphism and G is simply connected, there is a unique Lie group automorphism Λ t : G → G whose differential at the identity is λ t . For each t > 0, Λ t is a similarity with respect to the Carnot metric: d(Λ t (p), Λ t (q)) = t d(p, q) for any two points p, q ∈ G. For a Lie group isomorphism f : G → G ′ between two Carnot groups, the corresponding Lie algebra isomorphism f * :

The Baker-Campbell-Hausdorff formula
Let G be a simply connected nilpotent Lie group with Lie algebra g. The exponential map exp : g → G is a diffeomorphism. One can then pull back the group operation from G to get a group structure on g. This group structure can be described by the Baker-Campbell-Hausdorff formula (BCH formula in short), which expresses the product X * Y (X, Y ∈ g) in terms of the iterated Lie brackets of X and Y . The group operation in G will be denoted by ·. The pull-back group operation * on g is defined as follows. For X, Y ∈ g, define Then the first a few terms of the BCH formula ( [CG], page 11) is given by:

Pansu differentiability theorem
First the definition: Definition 2.1. Let G and G ′ be two Carnot groups endowed with Carnot metrics, and In this case, the strata preserving homomorphism L : G → G ′ is called the Pansu differential of F at x, and is denoted by dF (x).
The following result (except the terminology) is due to Pansu [P].
Theorem 2.1. Let G, G ′ be Carnot groups, and U ⊂ G, U ′ ⊂ G ′ open subsets. Let F : U → U ′ be a quasiconformal map. Then F is almost everywhere Pansu differentiable. Furthermore, at a.e. x ∈ U , the Pansu differential dF (x) : G → G ′ is a strata preserving isomorphism.

Fiber-preserving quasisymmetric maps
In this section we prove Theorem 1.1.
Let X, Y be (α, L)-fibered quasimetric spaces and F : X → Y an η-quasisymmetric map that sends fibers of X onto fibers of Y . We shall show that F is a (K, C)-quasi-similarity, where K depends only on η, α, L and the quasimetric constants of X, Y .
Lemma 3.1. If two fibers U and V in X are parallel, then F (U ) and F (V ) are parallel fibers in Y .
Proof. Suppose F (U ) and F (V ) are not parallel. By Condition (1.5), after possibly switching U and V , we may assume that there exist a sequence x i ∈ U such that d(F (x i ), F (V )) → ∞. Since U and V are parallel, for each i, there exists v i ∈ V such that d(x i , v i ) = d(U, V ). Since by Condition (1.2) the set U is path connected and unbounded, there are points on U at arbitrary distance from any point. Hence, for each i, there exists u i ∈ U such that d( The quasisymmetry condition and the fact d( Because of Condition (1.3), we can take a fiber U ′ = F (U ) parallel to F (U ). Again by  [PW]. The proofs are modifications of their arguments.
Lemma 3.2. There exists K 1 ≥ 1 depending only on η, α and L so that for any two parallel fibers U 1 , U 2 in X and any p, q ∈ U 1 satisfying d Proof. By Condition (1.2), there exists a geodesic space ( X 1 , d 1 ) and an L-biLipschitz map There is some integer k ≥ 2 such that The biLipschitz property of f 1 implies Since ( X 1 , d 1 ) is a geodesic space, there are points p = p 0 , p 1 , . . . , p k = q in U 1 such that Since U 1 and U 2 are parallel, Lemma 3.1 implies that F (U 1 ) and F (U 2 ) are also parallel. Let q i ∈ U 2 be a point such that d(F (p i ), F (q i )) = d(F (U 1 ), F (U 2 )). We have Since F is η-quasisymmetric, we have d(F (p i ), F (p i+1 )) ≤ η(L) · d(F (p i ), F (q i )) = η(L) · d(F (U 1 ), F (U 2 )).
By Condition (1.2) again, there exists some geodesic space ( Y 1 , ρ 1 ) and an L-biLipschitz map g 1 : (F (U 1 ), d) → ( Y 1 , ρ α 1 ). It follows that The triangle inequality for the metric space ( Y 1 , ρ 1 ) implies Now the biLipschitz property of g 1 implies Now the second inequality in (3.1) follows from (3.2) and (3.3). Finally, we notice that the second inequality for F −1 is equivalent to the first inequality for F .
Lemma 3.3. The following holds for any two distinct parallel fibers U 1 , U 2 and any two distinct p, q ∈ U 1 : where K 1 is the constant in Lemma 3.2.
Lemma 3.3 says that for any fiber U of X, the restriction F | U is a (K 3 1 , C)-quasisimilarity for some constant C > 0 that may depend on the fiber U . Next lemma states that this constant C can be chosen to be independent of the fiber.
Lemma 3.4. There is some constant C > 0 such that F | U is a (K 2 , C)-quasi-similarity for any fiber U of X, where K 2 is a constant depending only on η, α, L and the quasimetric constants of X, Y .
Proof. Fix a fiber U 0 (in X) and let U be an arbitrary fiber in X. By Lemma 3.3, there are constants C 0 , C > 0 such that F | U 0 is a (K 3 1 , C 0 )-quasi-similarity and F | U is a (K 3 1 , C)quasi-similarity. It suffices to show that there is a constant D depending only on η, α, L and the quasimetric constants of X, Y such that C 0 /D ≤ C ≤ DC 0 .
Let M ≥ 1 be a quasimetric constant for both X and Y . Fix x ∈ U and x 0 ∈ U 0 , and pick y ∈ U and y 0 ∈ U 0 such that The generalized triangle inequality in X applied to x, x 0 , y 0 implies The quasisymmetry condition now implies It follows that This together with the quasi-similarity property of F | U 0 and F Lemma 3.5. F is a (K, C)-quasi-similarity, where K depends only on η, α, L and the quasimetric constants of X, Y . .
Proof. Let p, q ∈ X be arbitrary. Let U, V be fibers in X such that p ∈ U , q ∈ V (we may have U = V ). Pick x ∈ U such that d(p, x) = d(p, q). Then So we have the upper bound for d(F (p), F (q)). The same argument applied to F −1 yields the lower bound for d(F (p), F (q)).
The proof of Theorem 1.1 is now complete.

An Example
Recall that two closed subsets U and V of a quasimetric space are defined to be parallel if d(u, V ) = d(v, U ) for any u ∈ U, v ∈ V . The following example shows that the conclusion of Theorem 1.1 fails if we replace "parallel" with the weaker condition HD(U, V ) < ∞.
Let X = Y = C be the complex plane with the usual metric. It is well known that for any α > −1, the map f α : C → C given by f α (z) = |z| α z is a quasisymmetric map. Let α > −1, α = 0, define a map F α : C → C as follows: Then it is clear that F α is a quasiconformal map, and hence is quasisymmetric. The fibers in X are horizontal lines, and the fibers in Y are images of horizontal lines under F α . A direct calculation shows that all fibers of Y are 10-biLipschitz to the real line. So Condition (1.2) is satisfied by both X and Y . All the conditions in Definition 1.1 are satisfied, provided we replace "parallel" with the weaker condition HD(U, V ) < ∞. However, F α is not biLipschitz due to the distorsion around the origin.

Quasisymmetric maps on Carnot groups with reducible first stratum
In this section we use Theorem 1.1 to prove Theorem 1.2. Specifically, we show that every quasisymmetric map on Carnot groups with reducible first stratum is biLipschitz. Let G be a Carnot group with Lie algebra g = V 1 ⊕ · · · ⊕ V s . Assume there is a nontrivial proper linear subspace W 1 ⊂ V 1 that is invariant under the action of the group of strata-preserving automorphisms of g. Let h be the subalgebra of g generated by W 1 . We denote the connected Lie subgroup of G with Lie algebra h by H and refer to it as the subgroup generated by W 1 . Notice that H is also a Carnot group and its lie algebra can be written as h = W 1 ⊕ · · · ⊕ W s . In general, there is some integer 1 ≤s ≤ s such that Ws = 0 and W j = 0 for j >s. Let ·, · be an inner product on V 1 and d the left-invariant sub-Riemannian Carnot metric on G determined by ·, · .
We recall the following: Proposition 4.1. (Proposition 3.4, [X2]) Let G and G ′ be two Carnot groups. Let W 1 and W ′ 1 be two subspaces of the first strata of the stratifications of Lie(G) and Lie(G ′ ), respectively. Let H and H ′ be the groups generated by W 1 and W ′ 1 , respectively. Let F : G → G ′ be a quasisymmetric map. If dF (x)(W 1 ) ⊂ W ′ 1 for a.e. x ∈ G, then F sends each left coset U of H into a left coset of H ′ . Now let F : G → G be an η-quasisymmetric map. By Pansu's differentiability theorem, at almost everywhere x ∈ G the map F is Pansu differentiable and the Pansu differential dF (x) : g → g is a strata-preserving automorphism. The assumption on W 1 in Theorem 1.2 implies that dF (x)(W 1 ) = W 1 . Now Proposition 4.1 implies that F sends left cosets of H to left cosets of H, where H is the subgroup generated by W 1 . Theorem 1.2 shall follow from Theorem 1.1 once we verify the conditions in Definition 1.1. Here the fibers are left cosets of H. Since H is also a Carnot group, (H, d) is a geodesic space. So Condition (1.2) is satisfied for α = L = 1. Conditions (1.3), (1.4) and (1.5) will be verified in Lemmas 4.2, 4.3 and 4.4, respectively.
Given a Lie algebra n and a subalgebra s ⊂ n, the normalizer of s in n is defined by: It is easy to see that N n (s) is a Lie subalgebra of n and s is an ideal in N n (s). Let N be a connected Lie group with Lie algebra n. Let S be the Lie subgroup of N with Lie algebra s. Let d be a left invariant distance on N . Lemmas 4.2 and 4.3 are valid for all connected and simply connected nilpotent Lie groups N with a left invariant distance. In particular, they hold true for the Carnot group G and left cosets of H. (4.1) Consequently, for any left coset U of S, there exist a sequence of left cosets U i of S that are parallel to U and converge to U .
Proof. For the first claim we induct on the degree of nilpotency. The claim holds when n is abelian since in this case N n (s) = n. Suppose the claim holds for all k-step nilpotent Lie algebras. Let n be (k + 1)-step. We assume N n (s) = s and will derive a contradiction from this. Notice the center C(n) of n lies in N n (s) = s. So we have a proper Lie subalgebra s/C(n) of the k-step nilpotent Lie algebra n/C(n). By the induction hypothesis, we have N n/C(n) (s/C(n)) = s/C(n). Let π : n → n/C(n) be the natural projection. It is easy to check that π −1 (N n/C(n) (s/C(n))) = N n (s). Since π −1 (s/C(n)) = s, we obtain N n (s) = s, contradicting the assumption N n (s) = s.
Regarding the second part of the lemma, let K be the connected subgroup of N with Lie algebra N n (s). Since s is an ideal in N n (s), we see that S is a normal subgroup of K. From (4.1) we deduce that K contains S properly. There exist k i ∈ K\S such that k i → e. So for any left coset U = gS, the left cosets U i = gk i S converge to U . Finally we claim that U i is parallel to U . Indeed, for any p = gs 0 ∈ U , where in the last equality we used the fact that Sk i = k i S.
Notice that the last part of the preceding proof shows that for any g ∈ N and any k ∈ K, the two left cosets gS and gkS are parallel.
Next lemma says that two different left costs of S can never get arbitrarily close. Proof. Set s 0 = s and define inductively s j = N n (s j−1 ). From (4.1) we have that s j properly contains s j−1 unless s j−1 = n. Since n is finite dimensional, there is some k such that s k−1 = n and s k = n. Let S j be the connected Lie subgroup of N with Lie algebra s j . Then S j−1 is a proper normal sugroup of S j for j ≤ k. It follows that for any g ∈ S j \ S j−1 , the two left cosets S j−1 and gS j−1 are parallel. Hence d(S j−1 , gS j−1 ) > 0. Now let U and V be two distinct left cosets of S. After applying a left translation we may assume U = S and V = gS for some g / ∈ S. There is some j ≤ k such that g ∈ S j \S j−1 . It follows that For the proof of the next lemma we will work under the assumption of Theorem 1.2.
Let ·, · be an inner product on g = V 1 ⊕ · · · ⊕ V s such that V i and V j are perpendicular to each other for i = j. Define a "quasi norm" on g as follows.
Next defined(x, y) = ||(−x) * y|| for x, y ∈ g. It is well known that every Carnot metric d on g is biLipschitz equivalent tod: there exists some constant C 0 ≥ 1 such that for any x, y ∈ g. Here we identified G and g via the exponential map. Proof. Suppose U and V are not parallel and HD(U, V ) < ∞. We shall obtain a contradiction. After applying a left translation we may assume U = H and V = gH for some g ∈ G\H. The last part of the proof of Lemma 4.2 shows that for any k in the normalizer K of H the two fibers H and kH = Hk are parallel. So g / ∈ K. Below we will identify g with G via the exponential map and do calculations in the Lie algebra using BCH formula. So we let Y / ∈ N g (h) be such that HD(h, Y * h) < ∞. Since we see that HD(h, Y * h * (−Y )) ≤ C for some constant C > 0.
We write Y = s i=1 Y i with Y i ∈ V i and let j ≥ 1 be the index such that Y j / ∈ N g (h) and Y i ∈ N g (h) for all i < j. By replacing Y with Y * (−Y 1 − . . . − Y j−1 ), we may assume Y i = 0 for i < j. Since [Y j , h] ⊂ h and h is generated by the first stratum W 1 , there is some (4.2) We next calculate Z := Y * (tX) * (−Y ) and A := (−X ′ ) * Z. We have where ad Y : g → g is the linear map given by ad and all the iterated brackets in the BCH formula for (−X ′ ) * Z are the sum of an element of h and an element in V j+2 ⊕ · · · ⊕ V s . So A j+1 is completely determined by −X ′ + Z. It follows that A j+1 is the sum of an element of W j+1 and t[Y j , X] (recall h = W 1 ⊕ · · · ⊕ W s ). Write [Y j , X] = B + B ⊥ , where B ∈ W j+1 and B ⊥ ∈ V j+1 is perpendicular to W j+1 with respect to the inner product <, > on g. Since [Y j , X] / ∈ h, we have B ⊥ = 0. Hence A j+1 equals the sum of tB ⊥ and an element of W j+1 . It follows that as t → ∞ since B ⊥ = 0. This contradicts (4.2).
We have verified all the conditions in Definition 1.1 and the proof of Theorem 1.2 is now complete.
Remark 4.3. Lemma 4.4 is equivalent to the following statement: if the Hausdorff distance between H and gHg −1 is finite, then g lies in the normalizer of H. We believe this is true for any left invariant distance on any connected, simply connected nilpotent Lie group N and any proper Lie subgroup H.

Other applications
In this subsection we give a couple of other applications of Theorem 1.1. The first is to quasiconformal maps of Heisenberg groups that send vertical lines to vertical lines. The second is to quasisymmetric maps of ideal boundary of certain amenable hyperbolic locally compact groups.
We first show that quasiconformal maps of the Heisenebrg groups H n = R 2n × R that permute vertical lines are biLipschitz. These maps are lifts of those biLipschitz maps of R 2n that preserve the standard sympletic form on R 2n . The reader is referrd to [T] and [BHT] for more details.
We next give a simple proof of Dymarz's theorem.
Theorem 4.6 ([D, Theorem 1]). Let N be a Carnot group equipped with a left invariant Carnot metric d and Q m (m ≥ 2) the m-adics with standard metric d m . Then every quasisymmetric map N × Q m → N × Q m is biLipschitz. Here the metric on N × Q m is given by d ′ ((x 1 , y 1 ), (x 2 , y 2 )) = max{d(x 1 , x 2 ), d m (y 1 , y 2 )}.
Recall that the standard metric d m on Q m is given by: where k is the smallest index for which a i = b i .
Proof. The fibers of N × Q m are the subsets N × {y} (y ∈ Q m ). Notice that (Q m , d m ) is perfect and totally disconnected. So the fibers of N × Q m are exactly the connected components of N × Q m . Hence every quasisymmetric map F of N × Q m permutes the fibers. Each fiber is isometric to a Carnot group and so is geodesic. Hence Condition (1.2) is satisfied for α = L = 1. The perfectness of Q m implies that parallel fibers are not isolated (Condition (1.3)). Distance between distinct fibers is clearly positive (Condition (1.4)). All fibers are parallel, so Condition (1.5) is vacuous. By Theorem 1.1 every quasisymmetric map of N × Q m is biLipschitz.