NETS and QUASI-ISOMETRIES 1. Basic Definitions and Extension

NETS AND QUASI-ISOMETRIES

D. BERNSTEIN, A. KATOK ∗, G.D. MOSTOW

This paper was written in the middle of 1980s and was accepted at that time by a leading mathematical journal modulo insigniﬁcant revisions. By essentially trivial reasons that did not involve either mathematical contents of the paper or any disagreement among the authors revised version was never submitted. Current publication is an initiative of the second author who takes full responsibility for any statements that may have become outdated and for possibly not citing more recent results that could be relevant to the material presented in the paper.

1. Basic definitions and extension lemma

Let X be a complete metric space with the distance function dX . Deﬁnition 1. A subset Γ ⊂ X is called a net if:

(1.1) Γ is uniformly discrete, i.e. there is r> 0 such that for γ1,γ2 ∈ Γ, γ1 6= γ2, dX (γ1,γ2) ≥ r (1.2) Γ spans X, i.e. there exists R > 0 such that for every x ∈ X one cane find γ ∈ Γ with dX (x, y) < R. The infimum of all R satisfying (1.2) will be called the spanning constant for Γ. Now let X, Y be two metric spaces. Definition 2. A map ϕ : X → Y is called a quasi-isometric embedding if there exist positive numbers A, B such that for any x1, x2 ∈ X, x1 6= x2: d (ϕx , ϕx ) A< Y 1 2

(1.3) AdX (x1, x2) − C < dY (ϕx1, ϕx2) < BdX (x1, x2)+ C Lemma 1. Let Γ be a net in a metric space X, ϕ : Γ → Y a pseudo-isometry. Then there exists a subset Γ′ ⊂ Γ such that Γ′ is also a net in X and the restriction of ϕ to Γ′ is a quasi-isometric embedding. This lemma is an almost immediate corollary of the following statement. Lemma 2. Let Γ be a net in a complete non-compact metric space X and T be a positive number. Then there exists a subset Γ′ ⊂ Γ which is also a net in X and such that: ′ (A) For every γ1,γ2 ∈ Γ , γ1 6= γ2, dX (γ1,γ2) >T . Proof. Let B(T ) be the collection of all subsets of Γ satisfying (A). It is non-empty and is partially ordered by inclusion. Any ordered subset of B(T ) has the maximal element (the union of all its elements); consequently by Zorn’s lemma, there is a maximal element Γ′ ∈ B(T ). This means

∗ This author is currently supported by NSF grants DMS 1002554 and 1304830. 1 2 D. BERNSTEIN, A. KATOK, G.D. MOSTOW

′ ′ ′ that for every point γ ∈ Γ there is a point γ ∈ Γ such that dX (γ,γ ) < T , because otherwise ′ ′ ′ Γ is not maximal. But then for every x ∈ X there is a point γ ∈ Γ such that dX (x, γ ) ≤ ′ ′ dX (x, γ)+ dX (γ,γ ) ≤ R + T , i.e. Γ is a net in X.

Proof of Lemma 1. If X is compact, let Γ′ be any one-point subset of Γ. If it is not compact, let us 2C ′ apply Lemma 2 with T = A . Then for any γ1,γ2 ∈ Γ one has:

A A A d (ϕγ , ϕγ ) > Ad (γ ,γ ) − C ≥ d (γ ,γ )+ T − C ≥ d (γ ,γ ) Y 1 2 X 1 2 2 X 1 2 2 2 X 1 2 and similarly since B ≥ A

3 B 3 d (ϕγ , ϕγ ) < Bd (γ ,γ )+ C < Bd (γ ,γ )+ C − T < Bd (γ ,γ ) Y 1 2 X 1 2 2 X 1 2 2 2 X 1 2 

Let us denote by Xn the nth cartesian power of the space X with the product topology. Further- more, let Σn be the standard (n − 1) simplex:

Σn = (t1,...,tn) : ti ≥ 0,i =1,...,n, ti =1 . ( i=1 ) X Deﬁnition 4. A centroid on a metric space X is a map

∞ n C : X × Σn → X n=1 [ satisfying the following properties (1.4)-(1.7): (1.4) For n = 1, C(x, 1) = x (1.5) C(x1,...,xn,t1,...,tk−1, 0,tk+1,...,tn) = C(x1,...,xk−1, xk+1,...,xn,t1,...,tk−1,tk+1,...,tn) n (1.6) C is continuous on every X × Σn (1.7) For every n and R> 0 there exists F (R,n) such that if dX (xi, xj ) ≥ R, i, j =1,...,n then dX (C(x1,...,xn,t1,...,tn), xi) < F (R,n) for every i =1,...,n and (t1,...,tn) ∈ Σn. It follows from (1.4) and (1.5) that:

C(x1,...,xn, 0,..., 0, 1, 0,..., 0) = xi (1.8) ↑ ith place

n k Definition 5. An n-centroid on X is a map defined on k=1 X × Σk and satisfying conditions (1.4)-(1.7). S n k Remark. Let us make the following identification on k=1 X × Σk: for each k and i identify (x1,...,xk,t1,...,ti−1, 0,ti+1,...,tk) with (x1,...,xi−1, xi+1,...,xk,t1,...,ti−1,ti+1,...,tk) and S call the corresponding topological space Xn. Then properties (1.5) and (1.6) can be expressed by n n k saying that there is a continuous map C : X → X such that C = C ◦ π, where π : k=1 X × Σk → Xn is the projection provided by the identification.b S A centroid C is called uniform if theb functionb F (R,n) in (1.7) canb be chosen independently of n. Theb following simple lemma is very useful for the construction of centroids. Lemma 3. Let X be a metric space in which balls of finite radius are compact. Then every 2- centroid on X can be extended to a centroid. NETS AND QUASI-ISOMETRIES 3

Proof. We will use induction in n, namely we set for n =3, 4,...

(1.9) C(x1,...,xn,t1,...,tn)= t1 tn−1 C C x1,...,xn−1, ,..., , xn,t1 + · · · + tn−1,tn t + · · · + t − t + · · · + t − 1 n 1 1 n 1 if at least one of the numbers t1,...,tn−1 is positive and

(1.10) C(x1,...,xn, 0, 0,..., 1) = xn. We assume that conditions (1.5)-(1.7) hold for k-centroids k =2,...,n − 1. Condition (1.5) for n-centroid follows immediately from (1.4) and (1.5) for 2-centroid. If k

C(x1,...,xn,t1,..., 0,...,tn)= t1 tn−1 = C C x1,...,xn−1, ,..., 0,..., , xn,t1 + · · · + tn−1,tn t + · · · + t − t + · · · + t − 1 n 1 1 n 1 t1 tk−1 = C C x1,...,xk−1, xk+1,...,xn−1, , , t + · · · + t − t + · · · + t − 1 n 1 1 n 1 tk+1 tn−1 ,..., , xn,t1 + · · · + tn−1,tn t + · · · + t − t + · · · + t − 1 n 1 1 n 1 = C(x1,...,xk−1, xk+1,...,xn,t1,...,tk−1,tk+1,...,tn)

Continuity of n-centroid at every point except of (x1,...,xn, 0,..., 0, 1) follows directly from (1.9) and from the continuity of (n − 1) and 2-centroids. In order to prove the continuity at (x1,...,xn, 0,..., 0, 1), let us notice that (n − 1)-centroid maps a set U × σn−1 where U is a neigh- borhood of (x1,...,xn) with compact closure into a compact closure into a compact set A. Since ′ ′ 2-centroid is uniformly continuous on compact sets we have uniformly for x ∈ A ′lim C(x, x ,t,t )= x →xn t→0 C(x, xn, 0, 1) = xn. In order to prove (1.7) for n-centroid, let us assume that we have found a function F (R,n − 1). We can also assume F (R, 2) and F (R,n − 1) are non-decreasing. We then have:

d(C(x ,...,x − ,t ,...,t − ), x ) ≤ d (x , x )+ (1.11) 1 n 1 1 n 1 n X n 1 dX (C(x1,...,xn−1,t1,...,tn−1), x1) ≤ R + F (R,n − 1) and from 1.9 for i =1,...,n − 1

dX (C(x1,...,xn,t1,...,tn), xi) ≤ ≤ dX (C(x1,...,xn,t1,...,tn), C(x1,...,xn−1,t1,...,tn−1)) + dX (C(x1,...,xn−1,t1,...,tn−1), xi) Using (1.7) for 2-centroid and (n − 1)-centroid and 1.11 we see that the ﬁrst term is estimated from above by F (F (R,n − 1)+ R, 2) and the second by F (R,n − 1). By the same reason

dX (C(x1,...,xn,t1,...,tn), xn) ≤ F (F (R,n − 1)+ R, 2). Thus we can put

(1.12) F (R,n)= F (F (R,n − 1) + R, 2)+ F (R,n − 1) and this function is non-decreasing 4 D. BERNSTEIN, A. KATOK, G.D. MOSTOW

Definition 6. We will call a metric space X uniformly locally compact if for any r, R the maximal number of points in any R-ball in X with pairwise distances ≥ r is bounded by a number depending only on R and r. Lemma 4 (Extension Lemma). Let us assume that a metric space X is uniformly locally compact and a space Y admits a centroid. Then every pseudo-isometry ϕ : Γ → Y of a net Γ ⊂ X into Y can be extended to a pseudo-isometry ϕ : X → Y with the same constants A and B and probably different C Proof. Let α be a continuous non-negativee function defined on the set of all positive numbers such that:

(1.13) α(t) → ∞ as t → 0 (1.14) α(t) > 0 for 0

Φα(x)= {γ ∈ Γ : α(dX (x, γ)) > 0} ∪ (Γ ∩{x}) .

By (1.2) and (1.14) the set Φα(x) is non-empty for every x. Since X is uniformly locally compact the number of elements in Φα(x) (which we denote by k(x)) is bounded from above by a number K.

We can represent Φα(x) in the following form γi1(x),γi2(x),...,γik(x)(x) , where i1(x) < i2(x) <

· · ·

(1.16) ϕ γij (x) = ϕj (x) and

α(dX (x, γi (x))) j , if x 6∈ Γ e k(x)   α(dX (x, γil (x))   l=1 (1.17) wj (x) =  X   0, if x ∈ Γ,γij (x) 6= x   1, if x ∈ Γ,γ = x  ij (x)  It follows from (1.3) and (1.15) that all the points ϕ (x) lie within at most 2(BT + C) from each  j other. We set now e

(1.18) ϕ(x)= C(ϕ1(x),..., ϕk(x)(x), w1(x),...,wk(x)(x)). The continuity of ϕ follows directly from (1.16), (1.17) and the properties of the centroid. Note that (1.8) guarantees thate ϕ is ane extensione of ϕ, and (1.1) and (1.13) provide continuity of ϕ at the points of Γ. e In order to check (1.3) lete us take arbitrary two points x1, x2 ∈ X and ﬁnd points γ1,γ2 ∈ Γ suche that dX (xi,γi) < R, i = 1, 2. Then all the points ϕj (xi), j = 1,...,k(xi) lie within the distance 2(BT + C) from ϕ(γi) and by (1.7) and (1.18) we have e dγ (ϕ(xi), ϕ(γi)) ≤ F (2(BT + C),K) and furthermore e NETS AND QUASI-ISOMETRIES 5

dγ (ϕ(x1), ϕ(x2)) ≤ dγ (ϕ(x1), ϕ(γ1)) + dγ (ϕ(x2), ϕ(γ2)) + dγ (ϕ(γ1), ϕ(γ2)) < BdX (γ1,γ2)+ C +2F (2(BT + C),K) ≤ B(dX (x1, x2)+2R)+ C +2F (2(BT + C),K) e e e ′ e = BdX (x1, x2)+ C ′ where C =2BR+2F ((2BT +C),K)+C. Similarly, we obtain dγ (ϕ(x1), ϕ(x2)) ≥ AdX (x1, x2)− ′ (2AR + C +2F (2(BT + C),K) ≥ AdX (x1, x2) − C e e 2. Examples of Centroid According to Lemma 3 in order to construct a centroid on a metric space N it is enough to deﬁne a continuous map C : N × N × [0, 1] → N such that

(2.1) C(x1, x2, 1) = x1, C(x1, x2, 0) = x2

and dN (x1, x2) < R implies

(2.2) dN (xi, C(x1, x2,t)) < F (R) We will show how to construct such a map in several important cases. Example 1. Let N be a complete Riemannian manifold such that every two of its points are connected by a unique geodesic.

Let us denote for x1, x2 ∈ N by Gx1,x2 the geodesic connecting x1 with x2 provided with the length parameter. Since the length of this geodesic is equal to the distance dN (x1, x2) we can represent Gx1,x2 as a map

Gx1,x2 : [0, dN (x1, x2)] → N where

(2.3) Gx1,x2 (0) = x1 and Gx1,x2 (dN (x1, x2)) = x2. Let us deﬁne a centroid C : N × N × [0, 1] → N by

(2.4) C(x1, x2,t)= Gx1,x2 (tdN (x1, x2)) Property (2.1) immediately follows from (2.3); (2.2) with F (R)= R folds because

dN (x1, Gx1,x2 (t)) + dN (x2, Gx1,x2 (t)) = dN (x1, x2)

Thus, it is left to prove the continuity of C. It is obviously continuous for x1 = x2. So let (n) (n) us assume that x1 6= x2, x1 → x1, x2 → x2, tn → t and G (n) (n) (tn) does not converge to x1 ,x2

Gx1,x2 (t). Since all the curves G (n) (n) lie in a compact part of N which can be covered by a ﬁxed x1 ,x2 number of coordinate charts one can use usual compactness argument in functional spaces to show that there is a subsequence of the sequence G (n) (n) which converges uniformly to a Lipschitz curve x1 ,x2

K : [0, dN (x1, x2)] → N diﬀerent from Gx1,x2 . It is easy to see that K(0) = x1, K(dN (x1, x2)) = x2. dN (x1,K(t)) = t so that the length of K is equal to dN (x1, x2). Thus, K must be a geodesic and by uniqueness K = Gx1,x2 . Let us point out two particular cases to which the above construction applies.

Example 1A. N is the universal covering of a compact Riemannian manifold of non-positive sectional curvature Example 1B. N = G/K where G is a connected semisimple Lie group, K its maximal compact 6 D. BERNSTEIN, A. KATOK, G.D. MOSTOW subgroup. Any left-invariant metric on G which is also right-invariant with respect ot K projects into a G-left invariant on N. Example 2. Let N be a connected Lie group of exponential type, i.e. the map exp : n → N from the lie algebra of N into N is one-to-one, provided with a left-invariant Riemannian metric. In this (x) case for every x ∈ N there is exactly one one-parameter subgroup gt of N such that x = g1(x). Let us deﬁne n o

−1 (2.5) C(x1, x2,t)= x1gt(x1 x2)

Since for every x ∈ N, g0(x) = e, g1(x) = x, condition (2.1) follows immediately from (2.5). Continuity is also obvious in this case because gt(x) depends continuously on both x and t. To verify (2.2) let us remark that since both the Riemannian metric on N and the centroid C are left-invariant one has

−1 C(x1, x2,t)= x1C(e, x1 x2,t) −1 where dN (e, x1 x2)= dN (x1, x2) and

−1 (2.6) dN (x1, C(x1, x2,t)) = dN (e, C(e, x x2,t)), −1 −1 (2.7) dN (x2, C(x1, x2,t)) = dN (x1 x2, C(e, x1 x2,t)).

But if dN (x1, x2) < R, all elements present in the right hand parts of (2.6) and (2.7) lie in the image of an R-ball in n under the exponential map. This image is a compact set and consequently is contained in a dN ball about e. We can choose the radius of that ball as F (R). Example 3. Examples 1B and 2 can be generalized in the following way. Let N be a metric space which is homeomorphic to Euclidean space Rm and which has a transitive group of isometries. Then N can be represented as I(N)/I0 where I(N) is the connected component of the identity in the group of isometries of N and I0 is the stabilizer of a point x0 in I(N). I(N) is a Lie group and I0 is its Lie subgroup so that I(N) is a locally trivial fibered bundle over N. The fiber over a point x ∈ N consists of all isometries from I(N) which map x into x0. Since N is contractible this fibered bundle is trivial, i.e. there is a continuous section ψ : N → I(N). Obviously, for x ∈ N

(2.8) ψ(x) · x = x0 m Let us ﬁx a homeomorphism Φ : R → N which maps the origin into x0. Let for x ∈ N

−1 Φ (x) = (s1(x),...,sm(x)) and for t ∈ R

gt(x) =Φ(ts1(x),...,tsm(x)) Obviously

(2.9) g0(x)= x0, g1(x)= x We are now ready to deﬁne a centroid:

−1 (2.10) C(x1, x2,t) = (ψ(x)) (g1−t(ψ(x1)x2)) Condition (2.1) follows directly from (2.10), (2.9) and (2.8). For,

−1 −1 C(x1, x2, 1)=(ψ(x1)) (g0(ψ(x1)x2)) = (ψ(x1)) x0 = x1 NETS AND QUASI-ISOMETRIES 7

and similarly

−1 −1 C(x1, x2, 0)=(ψ(x1)) (g1(ψ(x1)x2)) = (ψ(x1)) ψ(x1)x2 = x2 Continuity follows from the continuity of the section ψ and from (2.10). Finally, (2.2) can be proved as in Example 2. Namely,

−1 C(x1, x2,t) = (ψ(x1)) C(x0, ψ(x1)x2)

and since ψ(x1) is an isometry

dN (x0, ψ(x1)x2) = dN (x1, x2)

dN (x1, C(x1, x2,t)) = d(x0, C(x0, ψ(x1)x2,t))

dN (x2, C(x1, x2,t)) = d(ψ(x1)x2, C(x0, ψ(x1)x2,t))

But since Φ is a homeomorphism, the preimage of the R-ball around x0 in dN -metric is contained in a Euclidean ball of some radius, say T (R); the point C(x0, ψ(x1)x2,t) belongs to the image of that ball which is compact a consequently is contained in some ball in dN -metric. Let us denote the radius of this last ball by F (R). The most important particular case of the situation described above appears in the following context:

Example 3A. Let G be a connected Lie group, K ⊂ G - a maximal compact subgroup N = G/K - the homogeneous space provided with a metric invariant with respect to the left action of G. Then N is homeomorphic to a Euclidean space and consequently it admits a centroid.

3. Homotopy argument In this section we ﬁnd conditions which guarantee that in the situation described in Example 3, the extension described in Lemma 4 is a surjective map. Theorem 1. Let M, N be two complete metric spaces homeomorphic to Rm and Rn correspondingly. Let us assume that M has a transitive group of isometries. Let, furthermore, ϕ : M → N be a continuous map such that for every x, y ∈ M

dN (ϕ(x), ϕ(y)) ≥ AdM (x, y) − C for some constants A, C. Then (3.1) n ≥ m (3.2) If m = n, then ϕ(M)= N m Proof. Let us ﬁx a point x0 ∈ M, a homeomorphism Φ : M → R which maps x0 to the origin and a continuous section ψ : M → I(M) of the ﬁbered bundle I(M) → M. Let us consider in the Cartesian square M × M the “thickened diagonals” with respect to dM

δR = {(x, y) ∈ M × M : dM (x, y) ≤ R} and also Euclidean “thickened diagonals”

∆R = {(x, y) ∈ M × M : Φ(ψ(x)y) ∈ BR} m where BR is the Euclidean R-ball in R around the origin. Since Φ is a homeomorphism and ψ’s are isometries one can easily see that for every R> 0 one can ﬁnd f(R) such that

(3.3) ∆R ⊂ δf(R) 8 D. BERNSTEIN, A. KATOK, G.D. MOSTOW

and

(3.4) δR ⊂ ∆f(R)

The set MR = (M × M) \ ∆R for every R is a deformation retract of (M × M) \ ∆ = −1 −1 {(x, y) : x 6= y} via (x, y) → x, ψ(x) Φ (tΦψ(x)y)) and thus MR is homotopically equivalent to (M × M) \ ∆. Furthermore, the latter set is homotopically equivalent to the sphere Sm−1 = Rm m 1 (x1,...,xm) ∈ : i=1 xi =1 via the map P Φ(x) − Φ(y) (3.5) ν : (x, y) 7→ . M ||Φ(x) − Φ(y)||

Let σM : M × M → M × M be the standard involution given σM (x, y) = (y, x). Obviously, νM (σM z)= −νM (z). m−1 Lemma 5. Given any number R there exists a continuous map µ : S → MR such that

µ(−x)= σM (µx)

Proof. Since σM δR = δR, it follows from (3.3) and (3.4) that one can ﬁnd positive numbers R0 = R < R1 < · · · < Rm such that

(3.6) σM ∆Ri ⊂ ∆Ri+1 i =0,...,m − 1.

Let us take a point z ∈ MRm so that by (3.6) σM z ∈ MRm−1 . Since the set MRm−1 is homotopi- m−1 cally equivalent to S it is pathwise connected so that we can connect z and σM z in MRm−1 by a path λ0. By (3.6) σM λ0 ⊂ Mm−2. Clearly, the union λ0 ∪ σM λ0 deﬁnes a continuous map

1 µ1 : S → MRm−2 which is skew-symmetric, i.e.

µ1(−x)= σM µ1(x). i Continuing by induction in dimension and using (3.6) we can construct maps µi : S → MRm−i−1 for i =1,...,m − 1 such that

µi(−x)= σM µi(x)

For, assuming that µi−1 has been constructed and using the fact that πi−1(MRM−i )= 0, we can extend µi−1 to a continuous map

i λi−1 : S+ → MRm−i i i where S+ = (x1,...,xi+1) ∈ S : xi+1 ≥ 0 . Furthermore, since by (3.6) σM (λi−1(MRm−i )) ⊂ i M − we can extend λ − to a skew-symmetric continuous map µ : S → M − − by setting Rm i i 1 i Rm i 1 i λi−1(x) if x ∈ S+ µi(−x)= i i σM λi− (−x) if x ∈ S \ S 1 + Finally we can put µ = µm−1. 2C If the number R in Lemma 5 is chosen suﬃciently large, e.g. R>f A , where the function f comes from (3.3) and (3.4), then

(ϕ × ϕ)MR ⊂ (N × N) \ ∆ Consequently we can construct the composition map: NETS AND QUASI-ISOMETRIES 9

× m−1 µ ϕ ϕ νN n−1 (3.7) S −−−→ MR −−−→ (N × N) \ ∆ −−−−→ S

where νN is deﬁned similarly to (3.5). m−1 n−1 This composition map χ = µ(ϕ × ϕ)νN : S → S is skew-symmetric i.e. χ(−x)= −χ(x). By the Borsuk-Ulam theorem [1, p.337] such a map may exist only if m ≤ n and if m = n it is homotopically non-trivial. Consequently (ϕ × ϕ) = ϕ × ϕ| : M → (N × N) \ ∆ is also R MR R homotopically non-trivial.

Lemma 6. Given a point x0 ∈ M there is a homotopy ηt : MR → MR such that η0 = id, η1(MR) ⊂ M ×{x0}.

Proof. Let αt : M → M be the following standard homotopy between the identity and the map which sends all M to x0:

−1 αt(x)=Φ ((1 − t)Φ(x)) 0 ≤ t ≤ 1 Then let −1 ηt(x, y)= αtx, ψ(αtx) ψ(x)y −1 −1 Obviously, η0 = id and η1(x, y) = α1x, ψ(α1x) ψ(x)y = x0, ψ(x0) ψ(x)y = (x0, ψ(x)y) ∈ {x }× MR,x where 0 0 −1 MR,x0 = M \ Φ (BR)

We can include the restriction ϕ to MR,x0 into the following commutative diagram

ϕ MR,x0 −−−−→ N \{ϕ(x0)}

∗×id ∗×id

 (ϕ×ϕ)R  M −−−−−→ (N × N) \ ∆ yR y where ∗× id is the standard embedding: (∗× id)(x) = (x0, x) which is a homotopy equivalence according to Lemma 6. Thus, since (ϕ × ϕ)R is homotopically non-trivial, the map ϕ : MR,x0 → N \{ϕ(x0)} has the same property.

For every x ∈ MR,x0 we have from the assumption of the theorem dN (ϕ(x), ϕ(x0)) > AdM (x1, x2)− C > AR − C. In particular, if we ﬁx a point y0 ∈ N in advance, we can then choose R big enough so that y0 6∈ ϕ (MR,x0 ) and, moreover, there exists a homotopy βt : N → N identical on ϕ (MR,x0 ) and such that β0 = id and β1(ϕ(x0)) = y0.

This implies that ϕ : MR,x0 → N \{y0} is also nontrivial. On the other hand, since M is contractible, the map

ϕ : MR,x0 → ϕ(M) ⊂ N

is homotopically trivial which means that ϕ(M) can not miss the point y0.

4. Main Results Theorem 2. Let M = G/K, N = H/L be the factors of connected Lie groups G, H by their maximal compact subgroups K and L. Let us ﬁx on M and N a G and H-invariant Riemannian metrics correspondingly. Let Γ ⊂ M be a net with the spanning constant R and ϕ : Γ → N be a pseudo-isometry. Then (4.1) dim N ≥ dim M 10 D. BERNSTEIN, A. KATOK, G.D. MOSTOW

(4.2) ϕ can be extended to a continuous pseudo-isometry ϕ : M → N (4.3) If dim N = dim M, then ϕ(M) = N and the set ϕ(Γ) spans N with the spanning constant R1 ≤ BR + C where the constants B, C are determinede by the pseudo-isometry ϕ via (1.3). Proof. Since the space M admits ae transitive group of isometries it is uniformly locally compact (cf. Deﬁnition 6). The space N admits a centroid (cf. Example 3A). Thus, we can apply Lemma 4 and extend ϕ to a continuous pseudo-isometry ϕ : M → N. All assumptions of Theorem 1 hold for the map ϕ. Thus dim N ≥ dim M and if dim N = dim M then ϕ(M)= N there exists γ ∈ Γ such that dM (x, γ) < R. Consequently by (1.3) one hase e e dN (ϕ(γ),y)= dN (ϕ(γ), ϕ(x)) ≤ BdM (x, γ)+ C

(4.4) dG(g1,g2) − D ≤ dN (πg1,πg2) ≤ dG(g1,g2) These inequalities implie that both π and π ◦ ϕ : Γ → N are pseudo-isometries. By Lemma 1 there exists a next Γ′ ⊂ Γ in G such that the restriction of π to Γ′ is a quasi-isometric embedding. ′′ ′ ′′ By the same lemma there is another net Γ ⊂ Γ such that π ◦ ϕ|Γ′′ :Γ → N is a quasi-isometric embedding. Consequently πϕπ−1 : π(Γ′′) → N is also a quasi-isometric embedding, its image being π(ϕ(Γ′′)). Theorem 2 says then that π(ϕ(Γ′′)) spans N and by (4.4) ϕ(Γ′′) and hence ϕ(Γ), spans G.

5. Applications to Ergodic Theory Let (X, µ) be a Lebesgue measure space, i.e. a separable complete non-atomic probability measure space and let δ = {Sγ }γ∈Γ be a measurable right action of a locally compact second countable group Γ on X be non-singular transformations. Let G be another locally compact second countable group. A measurable function α : X × Γ → G is called a G=cocycle over the action g if for a.e. x ∈ X and for every γ1,γ2 ∈ Γ.

(5.1) α(x, γ2γ1)= α(x, γ1)α(δγ1 x, γ2) The construction of Mackey range [5], [3] Section 8, allows us to associate with any G cocycle α over a right Γ action a left action of G. Namely we ﬁrst determine a G-extension gα = Sα of γ γ∈Γ g which acts on X × G by

α (5.2) Sγ (x, g) = (δγ x, gα(x, γ)) NETS AND QUASI-ISOMETRIES 11

It is easy to see that the cocycle equation (5.1) is equivalent to the group property for that α α α extension Sγ1 Sγ2 = Sγ2γ1 .

The group G acts on X × G by the left shifts Lg0 : Lg0 (x, g) = (x, g0g) and this action obviously commutes with the extension Sα. In particular, this action maps orbits of Sα into orbits and thus we can consider the factor action of G in the space of orbits of Sα. This action is called Mackey range of α and is denoted by Lα. In general the space of Sα orbits may not have good measurable structure and event if it has such a structure and S is measure preserving, the natural Lα invariant measure may be infinite. We will give a sufficient condition which guarantees that the factor space has a structure of Lebesgue space with a natural finite invariant measure. Let us assume that Γ is finitely generated discrete group and that G is a locally compact Lie group. The word-length metrics on Γ determined by different systems of generators are equivalent in the sense that the identity map is a quasi-isometry. Similarly, all left-invariant Riemmanian metrics on G are equivalent. Thus the notions of quasi-isometric embedding and pseudo-isometry from Γ to G are intrinsically defined. Any left-invariant metric on Γ can be transferred to any orbit of a right Γ action. Thus, in our case the orbits of the action S are provided with a natural class of metrics defined up to a quasi-isometry. Definition 7. A G-cocycle α over a Γ action S is called a Lipschitz cocycle if for almost every x ∈ X the map αx :Γ → G, αx(γ)= α(x, γ) is a pseudo-isometry with constants A, B, C (cf. (1.3)) independent on x. Theorem 4. Let Γ be a uniform lattice in a connected Lie group G and let α be a Lipschitz G cocycle over a right measurable non-singular action S of Γ on a Lebesgue space (X, µ). Then the α Γ-action S has a measurable fundamental domain D = x∈X {x}× Dx where all sets Dx are uniformuly bounded and each Dx has a boundary of codimension one. Consequently if the action S S preserves the measures µ and if the group G is unimodular, the restriction of the measure µ × χG (χG is the Haar measure of G) to D determines a finite invariant measure for the Mackey range Lα.

Proof. It follows from the cocycle equation (5.1) that α(x, idΓ) = idG and

−1 −1 (5.3) α(x, γ )= α Sγ−1 ,γ

By Theorem 3 the set αx(Γ) for almost every x ∈ X spans G with a uniformly bounded spanning constant with respect to any left invariant Riemannian metric on G. The inversion g 7→ g−1 maps left-invariant metrics on G into right invariant ones. From now on we will work with a ﬁxed right- invariant Riemannian metric dG on G. Thus, by the above remark for almost every x ∈ X, the set −1 (αx(Γ)) spans G with respect to dG. −1 Let Dx(γ) be the Dirichlet region of the point (αx(γ)) , i.e.

−1 ′ −1 Dx(γ)= g ∈ G : dG g, α(x, γ) ≤ dG g, α(x, γ ) for all γ ∈ Γ −1 Since (αx(Γ)) is a discrete set, the boundary of each set Dx(γ) has codimension one. Theorem 3 guarantees that all sets Dx(γ) are compact and have bounded diameters. We obtain, using (5.1), (5.3) and the invariance of dG with respect to right multiplication on G, that Dx(γ) · α(x, γ) is exactly:

−1 −1 −1 ′ −1 ′ g ∈ G : dG g(α(x, γ)) , α(x, γ) ≤ dG (gα(x, γ) , (α(x, γ ) for all γ ∈ Γ ′ −1 ′ = g ∈ G : dG(g, id) ≤ dG(g, α(x, γ ) α(x, γ) for all γ ∈ Γ ′ − − ′ = g ∈ G : d (g, id) ≤ d (g, α(S x, γ γ 1)) 1 for all γ ∈ Γ G G γ = DS x (id) γ and by (5.2) 12 D. BERNSTEIN, A. KATOK, G.D. MOSTOW

α Sγ ({x}× Dx(γ)) = {Sγ x}× DSγx (id) Thus, every orbit of Sα visits the set

(5.4) D := {x}× Dx(id) x∈X [ at least once. If we assume that for every γ 6= idΓ, d(x, γ) 6= idG almost everywhere then the sets Dx(γ) form a partition of X × G up to a set of measure zero and the set defined by (5.4) is a fundamental domain for Sα satisfying all conditions of the theorem. In a general situation the Lipschitz condition from Definition 7 guarantees that the following equivalence relation has finite equivalence classes:

x ∼ y if y = Sγ x, α(x, γ) = id Moreover, the partition of X into equivalence classes is measurable. Let us choose a measurable set A ⊂ X which intersects each equivalence class by exactly one point and deﬁne

D = {x}× Dx(id) x∈A [ This set is obviously a fundamental domain for Sα and satisﬁes all conditions of the theorem. Remark 1. Construction used in the proof of Theorem 4 is a straightforward modiﬁcation of the construction from [2], Proposition 1, which deals with the case Γ = Zn, G = Rn. Our Theorem 3 replaces the ergodic theorem and elementary index arguments in Rn used in [2]. Another application of our results to ergodic theory involves the extension of the notion of Kakutani equivalence of group action to various classes of non-abelian groups. The construction is described in Section 8 of [3]. Details will appear in a separate paper.

References

[1] D.G. Bourgin, Modern algebraic geometry, McMillan, New York, 1963 [2] A. Katok, The special representation theorem for multi-dimensional group actions, Ast´erisque, 49, (1977), 117– 140 [3] A. Katok, Combinatorial constructions in ergodic theory and dynamics, American Mathematical Society, 2003. [4] G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, 1973. [5] R. J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis, 27, (1978), 350–372.