Quasi-flats in H^{2} × H^{2}

Alex Eskin and Benson Farb ^{∗} University of Chicago

April 24, 1997

Abstract

The purpose of this paper is to exhibit most of the main ideas of [EF] in the familiar special case of H^{2} × H^{2}. We prove here that any (coarse) quasiisometric embedding of R^{2} into H^{2} × H^{2} is close to a finite union of flats. The paper [EF] deals with the general case of a quasi-isometric embedding R^{n} → X for an arbitrary rank n symmetric space X of noncompact type.

1 Introduction

We recall the following:

Definition Let X and Y be metric spaces. A (κ, C) quasi-isometric embedding is a map ϕ : X ⟶ Y, such that

\frac{1}{κ}d_{X}(x_{1}, x_{2}) − C ≤ d_{Y} (ϕ(x_{1}), ϕ(x_{2})) ≤ κd_{X}(x_{1}, x_{2}) + C  (1)

Thus ϕ is bi-Lipschitz on large scales.

The main goal of this paper is to prove the following :

Theorem 1.1 (Quasi-flats are close to a finite union of flats). Let ϕ : R^{2} → H^{2} × H^{2} be a (κ, C)-quasi-isometric embedding. Then there are finite constants F = F (κ), N = N(κ, C) such that ϕ(R^{2}) lies in the N-neighborhood of a union of F flats in H^{2} × H^{2}.

∗Both authors are supported in part by N.S.F. Postdoctoral Fellowships. The work of the second author at MSRI was supported by NSF grant DMS-9022140.

In the paper [EF] we prove the analogous result for quasi-isometric embeddings R^{n} → X into an arbitrary nonpositively curved rank n symmetric space X with no Euclidean direct factors.

The purpose of the present paper is to exhibit most of the ideas used in the proof of the general result, but in a context that is easier to understand and to visualize. We refer the reader to [EF] for an introduction and motivation for the study of quasi-flats and quasi-isometries in symmetric spaces, as well as further applications.

1.1 Quasi-flats which are not close to flats

In this subsection we give an example, suggested to us by Bruce Kleiner, of a quasiisometric embedding of R^{2} in H^{2} × H^{2} which does not lie in a bounded neighborhood of a single flat in H^{2} × H^{2}. This type of example can be easily generalized to any symmetric space X with rank(X) ≥ 2.

Key Example: For each factor of H^{2} × H^{2}, pick three distinct geodesic rays in H^{2}, call them A, B, C and X, Y, Z. By taking metric products, each pair of rays gives rise to an isometrically embedded copy of a Euclidean quadrant in H^{2} × H^{2}. We denote these quarter-flats by AX, XB, etc. Divide the Euclidean plane R^{2} into six congruent sectors meeting at the origin. Let ϕ be the map which maps each sector, in clockwise order, to the quarter-flats AX, XB, BY, Y C, CZ, ZA by the quasi-isometry which maps a sector to a quadrant (see Figure 1).

Note that the image of any adjacent pair of sectors lies uniformly close to a halfflat in H^{2} × H^{2} since triangles in H^{2} are uniformly thin (see Figure 1). It follows that the restriction of ϕ to any pair of adjacent sectors is a (4, c)-quasi-isometry for some c, hence that ϕ is a (K, C)-quasi-isometry for some K, C. While ϕ(R^{2}) lies in a bounded neighborhood of a union of three flats, it clearly does not lie in a bounded neighborhood of a single flat.

Remarks.

1. The above construction can be carried out using any number of geodesic rays instead of just three. In this case the constant K of the quasi-isometry ϕ will get larger as the number of geodesic rays used increases.

2. Note that we may precompose the map ϕ : R^{2} → H^{2} × H^{2} with a self-quasi-isometry of R^{2}, such as a quasi-isometry taking rays to logarithmic spirals. The point is that the six sectors may be geometrically quite complicated. For this reason we will not be able to use the geometry of R^{2} in the proof of Theorem 1.1 Figure 1. The map ϕ : R^{2} → H^{2} × H^{2} maps each pair of adjacent sectors close to a half-flat: for example, the image of the shaded sectors under ϕ lies in a bounded neighborhood of the Euclidean half-flat γ × X in H^{2} × H^{2}, where γ is the geodesic connecting the endpoints of the rays A and B. It follows that ϕ is a quasi-isometry.

In what follows we always keep in mind the above examples.

1.2 Method of proof

In this section we give an outline of the proof of Theorem 1.1.

1.2.1 Nondegeneracy in X

In X = H^{2} × H^{2} there is a “degenerate subset”, which is simply the set of x ∈ X which project to the origin in one of the factors. For a fixed, very small tolerance δ > 0, we define the “nondegenerate subset” X^{δ} of X to be the set of x ∈ X with d(π_{i}(x), e) ≥ δd(x, e) for i = 1, 2, where π_{i} : X → H^{2} is the natural projection onto a factor and e ∈ X is the origin.

1.2.2 Coarse topology and pinched subsets

It is a well-known fact that there is no (κ, C)-quasi-isometric embedding of a large, equilateral, Euclidean triangle into H^{2}: if there were such a map, then the image would lie in a neighborhood of a geodesic triangle in H^{2} (since quasi-geodesics are close to geodesics), and triangles in H^{2} are uniformly thin (unlike equilateral triangles in R^{2}).

Intuitively, any subset of R^{2} which admits a (κ, C)-quasi-isometric embedding into H^{2} should not contain any fat loops. We formalize this in the Main Lemma (Lemma 4.7): If W is any subset of R^{2} which admits a (κ, C)-quasi-isometric embedding to a hyperbolic space, then for a certain (small) neighborhood W̃ of W, the natural homomorphism H_{1}(W) → H_{1}(W̃) induced by inclusion is the zero map. The size of the neighborhood W̃ of course depends on κ. We apply the Main Lemma to subsets of ϕ^{−1}(X^{δ})^{c}.

The coarse topology we use is in the spirit of [FS].

1.2.3 Outline of the proof of Theorem 1.1

Associated to X is its Furstenberg boundary X̂ = S^{1} × S^{1}, which is the product of the boundaries at infinity of the two H^{2} factors. Pairs of points in X̂ parameterize the set of (maximal) flats in X, just as pairs of points of the boundary at infinity of H^{2} parameterize the set of geodesics in H^{2}. We identify X̂ with the “visual sphere” S^{1} × S^{1} at the origin of X^{1}.

We consider the set of points S_{R} in R^{2} lying in the annulus A(R, 2R) which map into the nondegenerate subset X^{δ}, i.e. S_{R} = A(R, 2R) ∩ ϕ^{−1}(X^{δ}). We also consider the visual image of S_{R}, that is the set of points in the visual sphere S^{1} × S^{1} for which the infinite ray in that direction hits ϕ(S_{R}). In §3 we use basic geometry of X to show that the size of the visual image of ϕ(S_{R}) is small, in fact it can be covered by a bounded number (independent of R) of balls in S^{1} × S^{1} of size roughly e^{−R}.

Proving that the visual image of ϕ actually converges to a finite set of points (α_{1}, β_{1}), …, (α_{n}, β_{n}) in X̂ is a major step towards proving the theorem. Once this is known, we connect each α_{i} to each α_{j} by geodesics, and each β_{i} to each β_{j} by geodesics, and take all possible products of these geodesics to obtain a finite collection of flats. A final geometric argument, outlined in §6, proves that ϕ(R^{2}) lies close to this finite union of flats.

The main difficulty is that the visual images of the sets ϕ(S_{R}) for different R have no a priori relationship, for example the visual image of ϕ(S_{R_{j}}) may be a different finite collection of points for each R_{j} = 2^{j}. Hence there is no reason for the visual images of the sets ϕ(S_{R}) to converge to a fixed finite set of points in X̂.

We rule out this phenomenon by showing that visual images persist (cf. Lemma 4.9 and Proposition 4.10): Suppose that, for some sufficiently large R_{0}, some point x ∈ S_{R_{0}} maps into a “very nondegenerate subset” of X, i.e. ϕ(x) ∈ X^{b}, b ≫ δ, and denote by β ∈ S^{1} × S^{1} the visual position of ϕ(x). Then for every R ≥ R_{0} some point of ϕ(S_{R}) is seen in a direction near β. This fact together with the fact about the visual size of ϕ(S_{R}) noted above implies that there are finitely many limit points.

1Actually this is the product of the visual circles in each factor, as opposed to the visual sphere in X.

Proving persistence. A simple geometric observation (Lemma 3.3) shows that the visual size of any connected component of ϕ^{−1}(X^{δ}) is exponentially small. Thus, to prove visual images persist, it is enough to prove (Lemma 4.9) that every connected component of ϕ^{−1}(X^{δ}) which contains a “very nondegenerate point” x as above is unbounded.

The proof is simple in outline. First of all, since x maps into a very nondegenerate direction, the connected component of ϕ^{−1}(X^{δ}) containing x contains some large ball around x. If the lemma were not true then this component would be bounded, so could be completely surrounded by a set T ⊂ R^{2}, thought of as a circle S^{1}, which maps completely into one hyperbolic factor of X. By the coarse topology discussed above, this set T must be pinched, which contradicts the fact that it surrounds a large, round ball.

2 Preliminaries

2.1 Some definitions

Throughout the paper we let X = H^{2} × H^{2}. Both this and the rest of the notation are meant to match up with the notation in the general case of [EF], when X is any symmetric space. Let 0 ∈ R^{2} and e ∈ X be basepoints. Let B(0, r) denote the ball of radius r centered at 0, and let A(r_{1}, r_{2}) denote the annulus (centered at 0 unless otherwise specified) with inner radius r_{1} and outer radius r_{2}. Let ρ be a non-negative real number, and let d_{ρ}(x, 0) = max(d(x, 0), ρ). So d_{ρ} is the usual distance on R^{2}, except inside the ρ-ball all distances equal ρ.

For convenience we will assume that ϕ is continuous. It is possible to make this assumption by using the standard “connect the dots” argument, which changes ϕ into a continuous map while moving any image point by only a bounded amount. Connectthe-dots works roughly as follows: first triangulate the domain R^{2} by triangles of a uniformly bounded size; then build a quasi-isometry defined inductively on the skeleta of this triangulation, starting with the map being ϕ on the 0-skeleton, and extend to higher skeleta by connecting points (and then edges) by geodesics. This new quasiisometry is continuous, and lies a bounded distance (in the sup-norm) from ϕ.

Let K, ϵ > 0 be fixed. Unless otherwise stated, we will henceforth assume the following two conditions on the map ϕ, the point 0 ∈ R^{2}, and the point e ∈ X:

Condition I: ϕ is a “very coarse quasi-isometry” with additive coarseness constant depending on the distance from 0:

\frac{1}{K} d(x, y) − ϵd_{ρ}(x, 0) ≤ d(ϕ(x), ϕ(y)) ≤ Kd(x, y) + ϵd_{ρ}(x, 0)  (2)

where for simplicity of notation we assume here and throughout that d_{ρ}(x, 0) > d_{ρ}(y, 0).

Condition II: ϕ satisfies a radial condition:

\frac{1}{2K} d_{ρ}(x, 0) ≤ d(ϕ(x), e) ≤ 2Kd_{ρ}(x, 0)  (3)

Note. In all that follows we allow the conditions (2) and (3) to be violated inside some fixed ball B centered at the origin, as long as all points and paths we are considering are outside B. However, we never mention this point explicitly until §6.

Note that if ϕ is a (κ, C) quasi-isometry in the usual sense, then ϕ satisfies (2) with ρ = C/ϵ and K = κ. The constant ϵ can be made arbitrarily small by taking ρ sufficiently large. If ϕ(0) = e and ϵ is sufficiently small (depending on K), then (2) implies (3) outside some fixed ball.

The point of separating out the two conditions is that it will be necessary to consider the map ϕ with arbitrary points of R^{2} viewed as basepoints. The other reason to use Conditions I and II is to allow for maps ϕ that behave like quasi-isometries, but may remain undefined on many “holes” H ⊂ R^{2} of diameter roughly d(H, 0). Such maps occur naturally when considering noncocompact lattices in higher rank, such as SL(n, Z), n ≥ 3. The higher-rank analog of Theorem 1.1, or more precisely its extension to quasi-flats with holes (Theorem 7.1), plays a major role in the proof of quasi-isometric rigidity for such lattices (see [E]).

For any set U ⊂ R^{2}, we let U[c] be a neighborhood of U whose size is proportional to the distance from the origin, namely:

U[c] = {x ∈ R^{2} : ∃u ∈ U with d(x, u) < cd_{ρ}(x, 0).}

We use the same convention for subsets of X, with the origin replaced by the point e; namely, if U ⊂ X, then

U[c] = {x ∈ X : ∃u ∈ U with d(x, u) < cd(x, e).}

Let Y_{1} = H^{2} × {e} and Y_{2} = {e} × H^{2} be the direct factors of X, and denote the composition of ϕ with the natural projection π_{i} : X → Y_{i} by ϕ_{i}, i = 1, 2. For any δ > 0, we define

X^{δ} = X \ (Y_{1}[δ] ∪ Y_{2}[δ])

So the elements x ∈ X^{δ} are precisely those elements whose distance d(0, x) is comparable to the distance to the origin of each of its projections to the factors Y_{1}, Y_{2} (see Figure 2). More explicitly,

X^{δ} = {x ∈ X : d(π_{i}(x), e) ≥ δd(x, e) for each i = 1, 2}

Figure 2. What the intersections of Y_{i} and X^{δ} with a flat through e ∈ X look like in H^{2} × H^{2}.

Notational convention. All constants in this paper will depend on the quasiisometry constant K. Hence constants introduced in statements of lemmas and theorems without further mention should thought of as numerical constants which could, in principal, be computed from K and basic constants depending on X. We refrain from such explicit computations of constants only for clarity of presentation.

If a constant η further depends on constants a and b, we will simply write η = η(a, b). Unless otherwise specified, we denote by η_{j} “large” constants > 1, and by ν_{j} and δ_{j} “small constants”, > 0 and < 1. We use the notation O(b) to mean a quantity bounded above by λb for some implied constant λ = λ(K, X). We also write a ≪ b to mean a < νb for some small implied constant ν = ν(K, X) < 1, and a ≫ b to mean a > ηb for some large implied constant η = η(K, X) > 1. Finally we use the notations a ≺ b and a ≻ b to mean the same as a ≪ b and a ≫ b respectively, but without specifying the size of the implied constant.

We will always choose δ so that δ is much smaller then 1 but much larger then ϵ, i.e. ϵ ≪ δ ≪ 1, where ϵ is defined in Condition I.

2.2 Approximations

In this subsection we prove a few simple technical statements used throughout the paper.

2.2.1 Pullbacks and neighborhoods

Lemma 2.1. Let U ⊂ X be any subset. Then

(a) If ϵ ≪ δ ≪ 1 then ϕ^{−1}(U)[δ] ⊂ ϕ^{−1}(U[4K^{2}δ]).

(b) U[α][β] ⊂ U[α + β + αβ]. If α < 1 and β < 1, then U[α][β] ⊂ U[α + 2β].

(c) If α and β, ϵ ≪ α, β ≪ 1 then

ϕ^{−1} (U[α]) [β] ⊂ ϕ^{−1} (U[α + 4K^{2}β + 4K^{2}βα]) ⊂ ϕ^{−1} (U[α + 8K^{2}β]) .

In particular, if ϵ ≪ δ ≪ 1 then for i = 1, 2

ϕ^{−1}(Y_{i}[δ])[ϵ] ⊂ ϕ^{−1}(Y_{i}[2δ]).

Proof. Part (a) is an easy consequence of (2), (3), and the triangle inequality. Part (b) follows immediately from the triangle inequality. Part (c) follows immediately from combining Parts (a) and (b). The last statement is the case U = Y_{i}.

2.2.2 Approximating paths by edge-paths

For any ϵ > 0, we divide R^{2} into a grid G, that is a cellular decomposition of R^{2} into rectangular cells so that each point x ∈ R^{2} is contained in a cell each of whose edges has length between ϵd_{ρ}(x, 0)/2 and 2ϵd_{ρ}(x, 0). We emphasize that the size of the cells are not uniform: the length of an edge of a cell C is proportional to d_{ρ}(C, 0).

It will be necessary for us to approximate paths ϕ(γ) ⊂ X, γ an arbitrary path in R^{2}, by paths γ̂ which stay “close” to ϕ(γ), and so that there is some control on the length of γ̂.

Lemma 2.2 (Approximating paths by edge-paths). Let R > 0 be given, and let R_{j} = 2^{j}R for j = 1, 2 . . .. Let Â_{j} denote the annulus in X centered at e with inner radius R_{j} and outer radius R_{j+1}. Then for any path γ : [a, b] → R^{2}, there is a path γ̂ : [a, b] → X connecting (ϕ ∘ γ)(a) and (ϕ ∘ γ)(b) such that:

1. γ̂ ⊂ (ϕ ∘ γ)[λϵ], for some constant λ.

2. ℓ(γ̂ ∩ Â_{j}) ≺ \frac{1}{ϵ}R_{j} for every j = 1, 2, . . ..

Proof. Let C denote the union of cells C with C ∩ γ ≠ ∅. Then C is connected subcomplex of the grid G. Let a′ be the closest grid vertex to γ(a), b′ the closest grid vertex to γ(b). We define γ̃^{(ϵ)} to be the shortest edge-path in the 1-skeleton of C connecting a′ to b′, and let γ^{(ϵ)} = \overline{γ(a)a′} ∪ γ̃^{(ϵ)} ∪ \overline{b′γ(b)}, where \overline{γ(a)a′} and \overline{b′γ(b)} are segments in R^{2}. Then γ̂ is the path in X connecting ϕ ∘ γ(a) and ϕ ∘ γ(b) obtained by connecting the images under ϕ of the vertices of γ^{(ϵ)} by geodesics.

By definition of γ[2ϵ], and since every point of γ^{(ϵ)} lies in a cell intersecting γ and every cell C has edges of length at most 2ϵd_{ρ}(C, 0), we have γ^{(ϵ)} ⊂ γ[2ϵ].

Now for any z ∈ γ^{(ϵ)}, d(z, γ(t)) ≤ 2ϵd_{ρ}(z, 0) for some t. Hence

d(ϕ(z), ϕ(γ(t))) ≤ (2K + 1)ϵd_{ρ}(z, 0) ≤ (4K^{2} + 2K)ϵd(ϕ(z), 0)

the second inequality following from condition II. Hence ϕ(γ^{(ϵ)}) ⊂ ϕ(γ)[(4K^{2} +2K)ϵ]. By construction of γ̂ we now know γ̂ ⊂ ϕ(γ^{(ϵ)})[(2K + 1)ϵ]. Therefore by Lemma 2.1 part (b), γ̂ ⊂ ϕ(γ)[λϵ], with e.g. λ = 12K^{3}.

Now since γ^{(ϵ)} is shortest, it traverses any edge in C at most once. Hence

ℓ(γ^{(ϵ)} ∩ A(R_{j−1}, R_{j})) ≺ (# of edges in a cell) · (max. length of edge) · (# of cells)

≺ 2ϵR_{j} · \frac{(R_{j+1}^{2} − R_{j}^{2})}{(ϵR_{j}/2)^{2}}

≺ \frac{R_{j}}{ϵ}

By Condition II, ϕ^{−1}(Â_{j}) is contained in a bounded number of annuli A(R_{k−1}, R_{k}). Thus property 2 holds, by the definition of γ̂ and by Condition I. Note that γ^{(ϵ)} and γ̂ consist of geodesic segments, so that the inequality on the arc lengths follows from Condition I.

3 Visual Angles

3.1 A fact about H^{2}

For x, y ∈ H^{2}, let Θ_{e}(x, y) denote the visual angle between the points x and y when viewed from e, where e denotes the origin in H^{2}. We begin with an elementary but crucial lemma about the hyperbolic plane.

Lemma 3.1 (visual angle vs. hyperbolic distance in H^{2}). Let x, y ∈ H^{2} with d(x, e) = d(y, e) = r. If r is sufficiently large then :

(a) Any path from x to y staying outside the r-ball has length at least Θ_{e}(x, y)e^{r}/2.

(b) For any constant 0 < α < 1, if d(x, y) ≥ 2αr then Θ_{e}(x, y) ≥ e^{−(1−α)r}. (Taking α = 1/2 gives Θ_{e}(x, y) ≥ e^{−r/2} if d(x, y) ≥ r).

(c) For any constant 0 < α < 1, if Θ_{e}(x, y) ≥ e^{−αr} then d(x, y) ≥ 2(1 − α)r. (Taking α = 1/2 gives d(x, y) ≥ r if Θ_{e}(x, y) ≥ e^{−r/2}).

Proof. Now, the length of an arc on a circle of radius r is θ sinh r, where θ is the angle of the arc; part (a) follows by projecting the given path to the circle of radius r about the origin.

Parts (b) and (c) follows easily from the hyperbolic law of cosines.

3.2 Bounding the visual size of ϕ(R^{2}).

Let Θ(y) ∈ S^{1} denote the point such that exp(rΘ(y)) = y for some r > 0, where exp : T H_{e}^{2} → H^{2} denotes the exponential map. We recall the notation ϕ_{i} = π_{i} ∘ ϕ.

The following lemma says that the part of the ϕ-image of big annular regions in R^{2} which map into nondegenerate directions have small visual size when viewed from the origin in X.

Lemma 3.2 (bounded number of visual angles). There exists a constant ν_{1} = ν_{1}(δ) so that for each i = 1, 2, the set Θ(ϕ_{i}(A(R, 2R) ∩ ϕ^{−1}(X^{δ}))) can be covered by c_{0} = c_{0}(δ) balls of radius e^{−ν_{1}R} as long as R is sufficiently large (depending on δ, ρ).

In the Key Example (§1.1), for any R > 0 the set Θ(ϕ_{1}(A(R, 2R) ∩ ϕ^{−1}(X^{δ}))) consists of three points, corresponding to the directions of the three geodesic rays A,B, and C.

Proof of Lemma 3.2. For any R > 0, we decompose the unit circle S^{1} ⊂ T H_{e}^{2} into intervals of length at least e^{−ν_{1}R} and at most e^{−ν_{1}R/4}. We number the these intervals as I_{p}, p = 1, 2, . . ..

Suppose Θ(ϕ_{i}(x)) ∈ I_{p} and Θ(ϕ_{i}(y)) ∈ I_{p+2m} for m a nonzero integer. Then if ϕ(x) and ϕ(y) belong to X^{δ},

d(ϕ_{i}(x), e)) > δd(ϕ(x), e) > \frac{δR}{2K}

Similarly, d(ϕ_{i}(y), e) > δR/(2K). Since Θ_{e}(ϕ_{i}(x), ϕ_{i}(y)) ≥ e^{−ν_{1}R} by our choice of x and y, part (c) of Lemma 3.1 gives

d(ϕ_{i}(x), ϕ_{i}(y)) ≻ δR

for R sufficiently large. Then by Condition I, d(x, y) > λ_{2}δR/K for some constant λ_{2}.

From this it follows that the number of distinct m for which I_{p+2m} intersects Θ(ϕ_{i}(X^{δ}) ∩ A(R, 2R)) is bounded by (2R)^{2}/(λ_{2}δR/2K)^{2} = O(1/δ^{2}). Hence the the number of odd p (and even p) for which I_{p} ∩ Θ(ϕ_{i}(X^{δ}) ∩ A(R, 2R)) is non-empty is O(1/δ^{2}).

We emphasize that the size of the balls in Lemma 3.2 decreases exponentially with R, while the number of balls is bounded independent of R. However, Lemma 3.2 gives no information about the relative positions of the balls at different radii R.

The following lemma says that the visual size of the image of any connected component of ϕ^{−1}(X^{δ}) is small. In the lemma, and in the rest of the paper, connected means path-connected.

Lemma 3.3 (No Shifting Lemma). For every δ, ϵ ≪ δ ≪ 1, there exists ν_{2} = ν_{2}(δ) so that if x, y are any two points in the same connected component of ϕ^{−1}(X^{δ})∩ B(0, r)^{c}, then

Θ_{e}(ϕ_{i}(x), ϕ_{i}(y)) ≤ e^{−ν_{2} max(r,ρ)}

for each i = 1, 2 as long as max(r, ρ) is sufficiently large (depending on ϵ, δ).

In the Key Example (§1.1), a connected component of ϕ^{−1}(X^{δ}) consists of a Euclidean sector S of angle < π/3. The set Θ(ϕ_{i}(S)) ⊂ S^{1} is a single point, so in this case Lemma 3.3 is obviously true.

Proof of Lemma 3.3. Let R = max(r, ρ)/(2K). By Condition II, if d(x, 0) ≥ r and d(y, 0) ≥ r then d(ϕ(x), e) ≥ R and d(ϕ(y), e) ≥ R.

Let γ ⊂ ϕ^{−1}(X^{δ}) be a path such that γ(0) = x and γ(1) = y. Let γ̂ be the path constructed from ϕ ∘ γ as in Lemma 2.2. Since ϕ ∘ γ ⊂ X^{δ}, Property 1 of Lemma 2.2 gives γ̂ ⊂ X^{δ/2}, as long as ϵ ≪ δ. Let Â_{j} denote the annulus A(2^{j}R, 2^{j+1}R) in X. By Property 2 of Lemma 2.2 we have

ℓ(γ̂ ∩ Â_{j}) ≺ 2^{j}R/ϵ,  (4)

If ϕ(x) ∈ Â_{j} then d(ϕ(x), e) ≥ 2^{j}R. So if ϕ(x) ∈ Â_{j} ∩ X^{δ/2} then d(ϕ_{i}(x), e) ≥ δ2^{j−1}R for each i. Applying part (a) of Lemma 3.1 to the path π_{i}(γ̂) and with r = δ2^{j−1}R gives

|Θ(π_{i}(γ̂ ∩ Â_{j}))| ≤ 2ℓ(π_{i}(γ̂ ∩ Â_{j}))e^{−δ2^{j−1}R} by Lemma 3.1
≺ (2^{j+1}R/ϵ)e^{−δ2^{j−1}R}  by (4)

Summing over j then gives

|Θ(π_{i}(γ̂))| ≺ (R/ϵ) ∑_{j=0}^{∞} 2^{j+1}e^{−δ2^{j+1}R}  (5)

For sufficiently large r (and hence R), (5) is bounded by e^{−λδr} for some λ > 0, hence we can let ν_{2} = λδ.

4 Limit Sets

Let X̂ = S^{1} ×S^{1} be the product of the boundaries at infinity of the hyperbolic planes Y_{1} and Y_{2} in X. We call X̂ the boundary of X.

4.1 Definition and finiteness

Basic to our study is the definition of limit set of ϕ: the collection of points in the boundary X̂ on which ϕ(R^{2}) limits in a strong sense.

Definition 4.1 (Limit Point). A point β = (β_{1}, β_{2}) ∈ X̂ is called a δ-limit point of ϕ if there exists a path γ in ϕ^{−1}(X^{δ}) such that γ(t) leaves every compact set and the limit lim_{t→∞} Θ(ϕ_{i}(γ(t))) exists for each i and equals β_{i}. The collection of all δ-limit points of ϕ is called the δ-limit set of ϕ.

It follows from the main theorem that, for δ sufficiently small, the limit set is independent of δ. We will not give a direct proof of this assertion.

In the Key Example (§1.1), the limit set of ϕ consists of six pairs of points. For example, the point (a, x) ∈ X̂, where a (resp. x) is the endpoint at infinity of the ray A (resp. X) in H^{2}, is a limit point of ϕ.

Proposition 4.2 (The limit set is finite). The cardinality of the δ-limit set of ϕ is at most c_{0} (defined in Lemma 3.2).

Proof of Proposition 4.2. Suppose there exist c_{0} +1 limit points β_{1} = (β_{1}^{1}, β_{1}^{2}), …, β_{c_{0}+1} = (β_{c_{0}+1}^{1}, β_{c_{0}+1}^{2}). Since the β_{j} are limit points there exist paths γ_{j} : [0, ∞) → ϕ^{−1}(X^{δ}) which leave every compact set and lim_{t→∞} Θ(ϕ_{i}(γ_{j}(t))) = β_{j}^{i} for i = 1, 2. Choose r large so that Lemma 3.3 holds, so that for each i = 1, 2 the intervals of radius 2e^{−ν_{1}r} + 2e^{−ν_{2}r} centered at the limit points β_{1}^{i}, …, β_{c_{0}+1}^{i} in S^{1} are disjoint, and so that r > d(γ_{j}(0), 0) for all j. Here ν_{1} and ν_{2} are the same as in Lemma 3.2 and Lemma 3.3. For each j let t_{j} denote the greatest t such that d(γ_{j}(t_{j}), 0) = r, and let x_{j} = γ_{j}(t_{j}). Then by Lemma 3.3, Θ_{e}(ϕ_{i}(x_{j})), β_{j}^{i}) ≤ e^{−ν_{2}r} for i = 1, 2. Hence, by the triangle inequality, for p ≠ q, Θ_{e}(ϕ_{i}(x_{p}), ϕ_{i}(x_{q})) ≥ 2e^{−ν_{1}r} for i = 1, 2. Since there are c_{0} + 1 points x_{j}, this contradicts Lemma 3.2.

4.2 Independence of basepoint

In this subsection show that the δ-limit set of ϕ is independent of various choices of parameters.

For any point e′ ∈ X, we define X^{δ}(e′) the same as we defined X^{δ}, except now e′ is used as basepoint. More explicitly

X^{δ}(e′) = {x ∈ X : d(π_{i}(x), e′) ≥ δd(x, e′)) for each i = 1, 2}

Lemma 4.3. Suppose x, y ∈ X and let z be the midpoint of the geodesic connecting x and y. Let δ > 0 be such that y ∈ X^{δ}(x). Then x ∈ X^{δ}(y). Also x ∈ X^{δ}(z) and y ∈ X^{δ}(z).

Sketch of Proof. The first assertion follows immediately from the definitions, and the second is an easy similar triangles argument.

Lemma 4.4. Suppose x ∈ ϕ^{−1}(X^{δ}(e)), d(x, 0) ≥ R, and d(e, e′) ≤ r. Then x ∈ ϕ^{−1}(X^{δ/2}(e′)) if R ≫ r/δ.

Proof. This follows from Condition II and the triangle inequality.

The unit tangent circle T H_{z}^{2} at a point z ∈ H^{2} has a natural identification h_{z} : T H_{z}^{2} → S_{∞}^{1}, where a vector v⃗ ∈ T H_{z}^{2} maps to the endpoint of the unique geodesic ray through z in the v⃗ direction. The map h_{z} is a homeomorphism, indeed a conformal map. In this way the unit tangent circles at any two points z, y ∈ H^{2} may be identified in a natural way by the conformal homeomorphism h_{y}^{−1} ∘ h_{z}.

Lemma 4.5 (limit set is independent of basepoint). Suppose the map ϕ satisfies Condition I and Condition II with e ∈ X replaced by some other point e′ and 0 ∈ R^{2} replaced by 0′. Then the δ-limit set of ϕ defined at any e′ is contained the δ/2-limit set of ϕ defined at the origin e.

Proof. Let e′ ∈ X be any point, let r_{0} = d(e′, e) and let L′_{δ}(ϕ) denote the δ-limit set defined using e′ as basepoint. By Lemma 4.4, if β = (β_{1}, β_{2}) ∈ L_{δ}(ϕ) and γ is the path in ϕ^{−1}(X^{δ}) so that lim_{t→∞} Θ(ϕ_{i}(γ(t))) = β_{i}, then the part of γ lying outside a ball centered at e with radius ≫ r/δ is inside X^{δ/2}(e′). Also because hyperbolic triangles are thin, lim_{t→∞} Θ(gϕ(γ(t))) = g · β, where g is an isometry taking e to e′. Hence β ∈ L′_{δ/2}(ϕ) (recall that the isometry g^{−1} sending e′ to e also sends β to g^{−1} · β).

We also note the following:

Lemma 4.6 (invariance under linear growth perturbations). Suppose ϕ and ψ are two functions satisfying Condition I and Condition II, and d(ϕ(x), ψ(x)) ≤ ϵd_{ρ}(x, 0) for all x ∈ X. Then for ϵ ≪ δ ≪ 1, the δ-limit set of ϕ is contained in the (δ/2)-limit set of ψ and vice versa.

Proof. From part (c) of Lemma 2.1 ϕ^{−1}(X^{δ}) ⊂ ψ^{−1}(X^{δ/2}). Also by Lemma 3.1 part (c), Θ_{e}(ϕ_{i}(x), ψ_{i}(x)) → 0 as x → ∞ in ϕ^{−1}(X^{δ}) for i = 1, 2. Thus any δ limit point of ϕ is also a δ/2 limit point of ψ.

4.3 Existence of limit points

Our proof that limit points exist follows from the main lemma of this paper, whose proof we will leave for a later section. The idea is that the set of points in R^{2} which map (nearly) into one of the H^{2} factors of X must be a pinched subset of R^{2}. As stated in the introduction, our notion of W being pinched is the inclusion of W into some neighborhood inducing the zero map on first homology. Here we take the neighborhood of W to be W [ηδ], for some constant η. Since we wish to ignore effects inside a large (but fixed) ball, we union the sets in question with such a ball.

Lemma 4.7 (Main Lemma). Suppose ϵ ≪ δ ≪ 1 and max(r, ρ) is sufficiently large (depending on ϵ, δ). Then for any subset W of ϕ^{−1}(X^{δ})^{c}, the homomorphism i_{∗} : H_{1}(W ∪ B(0, r)) → H_{1}(W [ηδ] ∪ B(0, r)) induced by the inclusion is the zero map, for some η > 1 independent of δ.

In the Key Example (§1.1), the set ϕ^{−1}(X^{δ})^{c} consists of a neighborhood L[λ] (λ some constant depending on δ) of the union L of the six rays emanating from the origin in R^{2} (see Figure 2); in fact L = ϕ^{−1}(Y_{1} ∪ Y_{2}). Hence in this special example Lemma 4.7 is clearly true.

In the rest of this subsection we present consequences of the main lemma.

Lemma 4.8 (Non-Degenerate Points exist). If ϵ ≪ δ ≪ 1, and if R is sufficiently large (depending on ϵ, δ), then the sphere of radius R around 0 contains a point in ϕ^{−1}(X^{δ}).

Proof. Suppose r is so large that Lemma 4.7 holds. Let B = B(0, r) and let W denote the circle of radius R. If R > r, the sets W and B do not intersect. Also W [ηδ] ⊂ A((1 + ηδ)^{−1}R, (1 − ηδ)^{−1}R). Thus if R ≫ r, the sets W [ηδ] and B do not intersect. If the lemma is false, then W ⊂ ϕ^{−1}(X^{δ})^{c}. By Lemma 4.7, the inclusion induced homomorphism H_{1}(W ∪ B) → H_{1}(W [ηδ] ∪ B) is the zero map. Since W [ηδ] ∩ B = ∅, this implies that H_{1}(W) → H_{1}(W [ηδ]) is the zero map, and so H_{1}(W) → H_{1}(A((1 + ηδ)^{−1}R, (1 − ηδ)^{−1}R)) is also the zero map. This is clearly a contradiction since the inclusion W → A((1 + ηδ)^{−1}R, (1 − ηδ)^{−1}R) is that of a circle into an annulus.

The following lemma says that you can see forever after you see a deep point.

Lemma 4.9 (Connected components of ϕ^{−1}(X^{b}) are unbounded). Suppose ϵ ≪ b ≪ 1, and z is a point of ϕ^{−1}(X^{b}) with d_{ρ}(z, 0) sufficiently large (depending on b, ϵ). Then the connected component of ϕ^{−1}(X^{b/η_{1}}) ∩ B(0, d(z, 0))^{c} containing z is unbounded, for some η_{1} ≫ 1 independent of b.

Proof. All homologies and cohomologies in this proof are reduced with real coefficients. If ϵ ≪ δ ≪ 1, by Lemma 4.8 ϕ^{−1}(X^{b}) is unbounded. Pick δ so that ϵ ≪ δ and ηδ < b (here η is from Lemma 4.7). Let B denote the open ball B(0, d(z, 0)). Suppose the lemma is false, so that the component of ϕ^{−1}(X^{δ}) ∩ B^{c} containing z is bounded. Choose R so big that B(0, R − 1) contains this bounded component. Let W_{δ} = ϕ^{−1}(interior(X^{δ})) ∩ B(0, R + 1) and let W_{b} = ϕ^{−1}(interior(X^{b})) ∩ B(0, R). Since W_{b} ⊂ W_{δ}, W_{b} ∩ B^{c} has at least two connected components (one containing z and at least one intersecting the sphere of radius R − 1). Hence H_{0}(W_{b} ∩ B^{c}) ≠ 0. Furthermore the image of the map j_{∗} : H_{0}(W_{b} ∩ B^{c}) → H_{0}(W_{δ} ∩ B^{c}) induced by the inclusion is not zero. Note that W_{δ}^{c} = ⋃_{i=1,2} ϕ^{−1}(\overline{Y_{i}[δ]}) ∪ B(0, R + 1)^{c}. Therefore, by part (a) of Lemma 2.1, W_{δ}^{c}[ηδ] ⊂ W_{b}^{c}, since δ = b/η_{1} and η_{1} ≫ 1.

The universal coefficient theorem then implies that the corresponding map on cohomology j^{∗} : H^{0}(W_{δ} ∩ B^{c}) → H^{0}(W_{b} ∩ B^{c}) is not zero. Since W_{δ} and W_{b} are open, and W_{δ}^{c} ∪ B is orientable, we have the following commutative diagram:

H_{1}(W_{δ}^{c} ∪ B) ⟶ H_{1}(W_{b}^{c} ∪ B)
≅↓  ≅↓
H^{0}(W_{δ} ∩ B^{c}) ⟶ H^{0}(W_{b} ∩ B^{c})

The vertical maps are Alexander duality isomorphisms. Hence the top horizontal map is non-zero. But by the Main Lemma (Lemma 4.7) the top horizontal map is zero. This is a contradiction.

Proposition 4.10 (Limit Points Exist). Suppose ϵ ≪ b ≪ 1, and z is a point of ϕ^{−1}(X^{b}) with d_{ρ}(z, 0) sufficiently large (depending on b, ϵ). Then there exists a b/η_{1}-limit point β = (β_{1}, β_{2}) in X̂ such that

Θ_{e}(β_{i}, ϕ_{i}(z)) < e^{−ν_{2}d_{ρ}(z,0)} for i = 1, 2

The constant η_{1} is the same as in Lemma 4.9 and depends only on K; the constant ν_{2} depends on K and δ.

Proof. Let s = d_{ρ}(z, 0). By Lemma 4.9, there exists a path γ : [0, ∞) → ϕ^{−1}(X^{b/η_{1}}) ⊂ R^{2} such that γ(0) = z, d(γ(t), 0) ≥ s for all t and such γ leaves every ball centered at 0. Let r = max(ρ, s). For a non-negative integer j let t_{j} denote the greatest value of t for which d(γ(t_{j}), 0) = 2^{j}r. Then by Lemma 3.3, for all t ≥ t_{j}, Θ_{e}(ϕ_{i} ∘ γ(t), ϕ_{i} ∘ γ(t_{j})) ≤ e^{−ν_{2}2^{j}r} for i = 1, 2. This implies that the limit as t → ∞ of Θ(ϕ_{i} ∘ γ(t)) exists for each i: call it β_{i}. The above estimate for j = 0 shows that Θ_{e}(ϕ(z), β_{i}) ≤ e^{−ν_{2}r}.

5 Proof of the main lemma (Lemma 4.7)

5.1 The expanding annulus lemma

The following lemma is another statement of what it means for movement in H^{2} transverse to the radial direction to have “exponential cost”. Lemma 5.1 (Expanding annulus lemma in H^{2}). Let c_{1} ≪ 1 be given, and let ψ be a path in H^{2} connecting points a_{0} and b_{0}. Suppose that:

1. a_{0}, b_{0} ∈ A(r, (1 + c_{1}/4)r).

2. d(ψ(t), e) ≥ r for all t.

3. ℓ(ψ ∩ A(2^{i}r, 2^{i+1}r)) ≺ 2^{i}r. Then d(a_{0}, b_{0}) < c_{1}r if r is sufficiently large (depending on c_{1}). Proof. Let a′_{0} (resp. b′_{0}) denote the radial projection of a_{0} (resp. b_{0}) on a circle of radius r. Suppose d(a_{0}, b_{0}) ≥ c_{1}r. Then d(a′_{0}, b′_{0}) ≥ (c_{1}/2)r. Hence by part (b) of Lemma 3.1:

|Θ(ψ)| ≥ Θ_{e}(a_{0}, b_{0}) = Θ_{e}(a′_{0}, b′_{0}) ≻ e^{−r}  (6)

Part (a) of Lemma 3.1 gives

|Θ(ψ ∩ Â_{i})| ≺ ℓ(ψ ∩ A(2^{i}r, 2^{i+1}r))e^{−2^{j}r} ≺ (2^{j}r)e^{−2^{j}r}

Summing over j gives

|Θ(ψ)| ≺ r ∑_{j=0}^{∞} 2^{j}e^{−2^{j}r}

which contradicts (6) if r is large enough.

5.2 Coarse topology and pinched subsets of R^{2}

The following definition is motivated by Lemma 5.1.

Definition 5.2 (Pinching function). Let W be a subset of R^{2}. A non-negative, proper, continuous f : W → R is a pinching function on W, with pinching constants (r_{0}, η, β) if for all s ≥ r_{0} and all x, y ∈ W satisfying s ≤ f(x) ≤ f(y) ≤ ηs the following holds: if there exists a path γ : [0, 1] → W connecting x and y with f(γ(t)) ≥ s for all t, then d(x, y) < βs.

Note. If f is a pinching function for W then f is a pinching function for any subset of W, with the same pinching constants.

The goal of this subsection is to show that subsets W ⊂ R^{2} with pinching functions are pinched, by which we mean the homology H_{q}(W), q ≥ 1 vanishes under the inclusion map into some neighborhood of W.

We begin with a fact about R^{2}.

Lemma 5.3 (Trivial R^{2} fact). If r > 0 and if T ⊂ R^{2} is a set of diameter at most r, then for any function r(x) on T satisfying 2r < r(x) < 4r for all x ∈ T, the set ⋃_{x∈T} B(x, r(x)) is contractible.

Proof. Let u be any point of T. Then T ⊂ B(u, r). Hence for each x ∈ T, the ball B(x, r(x)) contains u. Thus ⋃_{x∈T} B(x, r(x)) is star-shaped with respect to u, hence contractible.

Lemma 5.4 (Pinching lemma). Suppose W ⊂ R^{2}, and suppose f is pinching function on W with pinching constants (r_{0}, η, β). Let F̃ be any ball containing f^{−1}([0, r_{0}]), and let W̃ be the following the neighborhood of W:

W̃ = F̃ ∪ {x ∈ R^{2} : ∃w ∈ W with d(x, w) < 4βf(w)}

Then the homomorphism i_{∗} : H_{1}(W) → H_{1}(W̃) induced by the inclusion i : W → W̃ is the zero homomorphism.

Proof. Let F = W ∩ F̃, so F̃ contains F and W̃ is a neighborhood of W. From the commutative diagram

\begin{matrix}H_{1}(F) & ⟶ & H_{1}(W) & ⟶ & H_{1}(W, F) \\ ↓ & & ↓ & & ↓ \\ H_{1}(F̃) & ⟶ & H_{1}(W̃) & ⟶ & H_{1}(W̃ , F̃) \end{matrix}  (7)

and the contractibility of F̃, it is enough to show that the right vertical homomorphism in (7) is the zero map.

In the proof of this lemma, for any subset Q of W, we denote by Q̃ the union of the set F̃ with the set of points x ∈ R^{2} such that there exists w ∈ Q with d(x, w) < 4βf(w).

Clearly, without loss of generality we may assume that W is bounded. (Indeed, if c is a cycle such that i_{∗}(c) ≠ 0, we may replace W by a suitable metric neighborhood of c.) Pick a decreasing sequence s_{0}, …, s_{N} so that s_{0} = sup_{w∈W} f(w), s_{k−1}/s_{k} < η^{1/2} for all k, and s_{N} = r_{0}. Let W_{k} = {w ∈ W : f(w) ≤ s_{k}} ∪ F, so that W_{k} is a decreasing family of sets with W_{0} = W, and W_{N} = F.

Figure 3. Applying the definition of pinching function shows that each of the regions like the shaded regions must have small diameter. Hence W = ∪_{i=1}^{N}W_{i} must

be pinched.

We claim that for each k, and any connected component Q of W \ W_{k+1},

j_{∗} : H_{1}(Q, Q ∩ W_{k}) → H_{1}(Q̃, { \atop Q ∩ W_{k}}) is the zero map (*)_{k}

where j_{∗} is induced by the inclusion j : Q → Q̃. The point of considering the connected components as opposed to all of W \ W_{k} at once is that this allows us to get the additional property of connectedness, which will be crucial.

We prove (∗)_{k} by induction on k. If k = 0 then H_{1}(Q, W_{0}) = 0 because Q ⊂ W = W_{0}. Now suppose (*)_{k−1} is true. Consider the commutative diagram:

\begin{matrix}H_{1}(Q ∩ W_{k−1}, Q ∩ W_{k}) & ⟶ & H_{1}(Q, Q ∩ W_{k}) & ⟶ & H_{1}(Q, Q ∩ W_{k−1}) \\ ↓ & & ^{j_{∗}}↓ & & ^{j′_{∗}}↓ \\ H_{1}({ \atop Q ∩ W_{k−1}}, { \atop Q ∩ W_{k}}) & ⟶ & H_{1}(Q̃, { \atop Q ∩ W_{k}}) & ⟶^{Φ_{∗}} & H_{1}(Q̃, { \atop Q ∩ W_{k−1}}) \end{matrix} (8)

where the top row is the exact sequence of the triple Q ∩ W_{k} ⊂ Q ∩ W_{k−1} ⊂ Q, the bottom row is the exact sequence of the triple { \atop Q ∩ W_{k}} ⊂ { \atop Q ∩ W_{k−1}} ⊂ Q̃, and the vertical maps are induced by the inclusion Q → Q̃.

The fact that j′_{∗} in (8) is zero does not formally follow from the inductive hypothesis (*)_{k−1} because Q is a connected component of W \ W_{k+1}, not W \ W_{k}. However, we may argue as follows: since W_{k} ⊂ W_{k−1} for all k,

H_{1}(Q, Q ∩ W_{k−1}) = H_{1}(Q \ W_{k}, (Q \ W_{k}) ∩ W_{k−1}) = ⊕_{i∈I}H_{1}(S_{i}, S_{i} ∩ W_{k−1})

where the S_{i} are the connected components of Q \ W_{k}. The inductive hypothesis (∗)_{k−1} is the statement that for each i, the homomorphism j_{∗} : H_{1}(S_{i}, S_{i} ∩ W_{k−1}) → H_{1}(S̃_{i}, { \atop S_{i} ∩ W_{k−1}}) induced by the inclusion S_{i} → S̃_{i} is the zero map. Since for each i the inclusion S_{i} → Q̃ factors through the inclusion S_{i} → S̃_{i}, this implies that the map j′_{∗} in (8) is the zero map.

Denote (Q ∩ W_{k−1}) \ (Q ∩ W_{k}) by T. Now suppose x and y are any two points in T. Then s_{k} ≤ f(x) ≤ s_{k−1}, and s_{k} ≤ f(y) ≤ s_{k−1}. By the definition of T, and since Q is connected, x and y can be connected by a path γ : [0, 1] → W such that f(γ(t)) ≥ s_{k+1} for all t. Note that s_{k−1}/s_{k+1} < η. Since f is a pinching function, this implies that d(x, y) < βs_{k+1}. Now Lemma 5.3 implies that T̃ is contractible. Hence H_{1}({ \atop Q ∩ W_{k−1}}, { \atop Q ∩ W_{k}}) = 0, so the map Φ_{∗} in (8) is injective. This, together with the fact that j′_{∗} = 0, implies that the map j_{∗} in (8) is the zero map, which proves (*)_{k}. Since W_{N} = F, this proves that the right vertical homomorphism in (7) is indeed zero.

5.3 Constructing a pinching function on ϕ^{−1}(Y_{i}[δ])

5.3.1 An approximation

Lemma 5.5 (ϕ_{i} satisfies Conditions I and II). Let ϵ ≪ δ ≪ 1. For each i = 1, 2, the following two facts hold for any x, y ∈ ϕ^{−1}(Y_{i}[δ]) with d(x, 0) ≥ d(y, 0):

d(x, y) − O(δd_{ρ}(x, 0)) ≺ d(ϕ_{i}(x), ϕ_{i}(y)) ≺ d(x, y) + O(δd_{ρ}(x, 0))  (9)

2.

\frac{1}{4K} d_{ρ}(x, 0) ≤ d(ϕ_{i}(x), e) ≤ 4Kd_{ρ}(x, 0)  (10)

Proof. If x, y ∈ ϕ^{−1}(Y_{i}[δ]) then ϕ(x), ϕ(y) ⊂ Y_{i}[δ]. Note that on Y_{i}[δ],

d(x, π_{i}(x)) = O(δd(x, e)).  (11)

Hence

d(ϕ(x), ϕ_{i}(x)) = O(δd(ϕ(x), e)) = O(δd_{ρ}(x, 0))  (12)

by Condition II. Applying the triangle inequality together with (12) and (2) then gives (9).

To prove the second fact, we know from (12) and Condition II that, if ϵ ≪ δ,

\frac{1}{2K} d_{ρ}(x, 0) − O(δd_{ρ}(x, 0)) ≤ d(ϕ_{i}(x), e) ≤ 2Kd_{ρ}(x, 0) + O(δd_{ρ}(x, 0)) (13)

Now (10) follows from (13) if δ ≪ 1.

5.3.2 Constructing a pinching function

The motivation for the definition of pinching function came from the behavior of functions such as d(g(x), e), where g is a reasonably efficient map of R^{2} into H^{2}, such as ϕ_{i} restricted to ϕ^{−1}((X^{δ})^{c}). The property of H^{2} used was that “travel transverse to the radial direction in H^{2} is exponentially expensive”, as given precisely by the Expanding Annulus Lemma (Lemma 5.1).

The pinching functions we need to construct, however, use instead the neighborhood Y_{i}[δ] of a hyperbolic plane. This neighborhood only introduces a small linear error term O(δr) in the functions involved, so proving that d(ϕ_{i}(x), e) is a pinching function on ϕ^{−1}(Y_{i}[δ]) should clearly follow from the Expanding Annulus Lemma. However, one must keep track of the error term and show that the axioms of a pinching function are satisfied.

Lemma 5.6 (Constructing a pinching function on ϕ^{−1}(Y_{i}[δ])). If ϵ ≪ δ ≪ 1, then for each i = 1, 2 the set ϕ^{−1}(Y_{i}[δ]) has a pinching function f with pinching constants (r_{0}, ω = 1 + O(δ), β = O(δ)), where r_{0} = r_{0}(δ). Furthermore, f also satisfies:

\frac{1}{2K} d_{ρ}(x, 0) ≤ f(x) ≤ 2Kd_{ρ}(x, 0) for all x ∈ ϕ^{−1}(Y_{i}[δ])  (14)

Proof. We claim that f(x) = d(ϕ_{i}(x), e) is a pinching function on ϕ^{−1}(Y_{i}[δ]). Equation (14) follows directly from Condition II and the fact that the domain of f lies in ϕ^{−1}(Y_{i}[δ]). This also shows that f is proper.

Now suppose x, y ∈ ϕ^{−1}(Y_{i}[δ]), s ≥ r_{0} and s ≤ f(x) ≤ f(y) ≤ ωs. Let γ : [0, 1] → ϕ^{−1}(Y_{i}[δ]) be a path from x to y satisfying f(γ(t)) ≥ s for all t. We must show d(x, y) ≺ δs.

Let γ̂ be the path constructed from ϕ ∘ γ as in Lemma 2.2. Property 1 of Lemma 2.2 gives γ̂ ⊂ (ϕ ∘ γ)[λϵ]. Hence γ̂ ⊂ Y_{i}[2δ] since ϵ ≪ δ. Also the fact that γ̂ ⊂ (ϕ ∘ γ)[λϵ] implies that for every point x̂ ∈ γ̂ there is a point ŷ ∈ ϕ(γ) so that d(x̂, ŷ) = O(ϵd(ŷ, e)). Hence, if ϵ ≪ δ ≪ 1,

d(π_{i}(ŷ), π_{i}(x̂)) ≺ δd(ŷ, e) ≺ δd(π_{i}(ŷ), e)

Thus

d(π_{i}(x̂), e) ≥ d(π_{i}(ŷ), e) − d(π_{i}(ŷ), π_{i}(x̂)) = (1 − O(δ))d(π_{i}(ŷ), e).

Since ŷ = ϕ ∘ γ(t) for some t ∈ [0, 1], d(π_{i}(ŷ), e) = f(γ(t)) ≥ s. Thus for all x̂ in the image of γ̂,

(1 − λδ)s ≤ d(π_{i}(x̂), e) ≤ d(x̂, e)  (15)

for some constant λ.

We want to apply Lemma 5.1 to the path ψ = π_{i} ∘ γ̂. Let r = (1 − λδ)s. Then by (15), d(ψ(t), e) ≥ r for all t in [0, 1], hence the first hypothesis of Lemma 5.1 holds.

Let Â_{j} denote the annulus A(2^{j}r, 2^{j+1}r) in H^{2}. It follows that π_{i}^{−1}(Â_{j}) ∩ Y_{i}[2δ] lies in an annulus A(2^{j−2}r, 2^{j+2}r) in X. By Property (2) of Lemma 2.2, we then get the bound

ℓ(π_{i} ∘ γ̂) ∩ Â_{j} ≺ 2^{j+3}r/ϵ

so that hypothesis (3) in Lemma 5.1 holds.

Let c_{1} = 8λδ, ω = 1 + (λ/2)δ, a_{0} = ϕ_{i}(x), and b_{0} = ϕ_{i}(y). Since d(a_{0}, e), d(b_{0}, e) ∈ [s, ωs], it easy to check that our choice of c_{1} and ω gives d(a_{0}, e), d(b_{0}, e) ∈ [r, (1 + c_{1}/4)r], as long as δ ≪ 1. So we may apply Lemma 5.1, which says that if r is sufficiently large then

d(ϕ_{i}(x), ϕ_{i}(y)) = d(a_{0}, b_{0}) < c_{1}r < c_{1}s = O(δs)

Hence

d(x, y) ≺ c_{1}s + O(δd_{ρ}(x, 0)) by (9)
≺ c_{1}s + O(δd(ϕ_{i}(x), e) by (10)
≺ c_{1}s + O(δs)  since d(ϕ_{i}(x), e) ≤ ωs ≤ 2s
≺ δs

5.4 Proof of the Main Lemma (Lemma 4.7)

Proof of Lemma 4.7. Since if f is a pinching function on a set S, then f is a pinching function on any subset of S, and the first assertion follows immediately from Lemma 5.6 and Lemma 5.4, and the observation that by (14) and the definition of the pinching constant β, W̃ ⊂ W [ηδ] for η = O(1).

6 Proof of Theorem 1.1

We will need the following lemma about triangles in hyperbolic space.

Lemma 6.1 (lemma about triangles in H^{2}). For every ν > 0 there exists Λ > 0 so that if γ is a geodesic in H^{2} with endpoints x_{1}, x_{2} and Θ_{p}(x_{1}, x_{2}) > ν. then d(p, γ) < Λ.

Proof. This follows easily from the hyperbolic law of cosines.

There are simpler proofs of Theorem 1.1 than the one we are about to give. We give the following proof because it is the one that can be generalized to other higher rank symmetric spaces X (see [EF]).

We are given a (κ, C)-quasi-isometric embedding ϕ : R^{2} → X. The idea of the proof is roughly as follows.

First off, suppose z ∈ X is a point which sees two far-away points ϕ(x_{1}), ϕ(x_{2}) ∈ X^{δ}(z) with both visual angles Θ_{π_{i}(z)}(ϕ_{i}(x_{1}), ϕ_{i}(x_{2})) bounded below. By Lemma 4.9, for each j = 1, 2, there is a path γ_{j} from x_{j} to infinity in ϕ^{−1}(X^{δ}(z)) with ϕ ∘ γ_{i} limiting to a limit point β_{j} ∈ S^{1} × S^{1}. If we knew that the paths ϕ ∘ γ_{j} stayed outside a large ball containing z, then by the Lemma 3.3 (no shifting lemma) β_{1} ≠ β_{2}. Since the visual distances Θ_{π_{i}(z)}(ϕ_{i}(x_{1}), ϕ_{i}(x_{2})) are bounded below, the visual distance Θ_{π_{i}(z)}(π_{i}(β_{1}), π_{i}(β_{2})) is bounded below (again by no shifting). Then by Lemma 6.1 π_{i}(z) must be a bounded distance from the geodesic λ_{i} connecting π_{i}(β_{1}) to π_{i}(β_{2}). Let F (β_{1}, β_{2}) = λ_{1} × λ_{2} be the flat which is the product of these geodesics. It is then clear that z is a bounded distance from the flat F (β_{1}, β_{2}).

Why do the paths ϕ ∘ γ_{j} stay outside a large ball containing z? There are two cases: either z is far from ϕ(R^{2}) (and we are obviously done), or ϕ(R^{2}) comes close to z at a point p ∈ ϕ(R^{2}), in which case we can (Lemma 4.9) find paths γ_{i} as above which stay far from ϕ^{−1}(p). Then ϕ ∘ γ_{j} would stay far from p since ϕ is a quasiisometry. Since p and z are close, this implies that the paths stay far away from z as well.

Now given an arbitrary x_{1} ∈ R^{2}, we can use Lemma 4.8 to find x_{2} ∈ R^{2} which is a large (but fixed) distance R from x_{1}, and so that each ϕ(x_{i}) ∈ X^{δ}(z), where z is the midpoint of the geodesic from ϕ(x_{1}) to ϕ(x_{2}). We then apply the above argument to show that z is close to a flat connecting two limit points of ϕ, and so ϕ(x_{1}) is close plus κR/2 to that flat.

Notice that, in the above argument, we really had to apply all of the previous machinery that gave existence of limit points two different times. We write out the step which will be repeated in the following:

Proposition 6.2. Suppose 0, x_{1}, x_{2} ∈ R^{2}, e ∈ X and ϵ, δ, K, λ_{0}, r > 0 are such that

(a) ϕ satisfies (2) and (3) outside some ball B centered at 0.

(b) ϵ ≪ δ ≪ 1 (the implied constant depending on K).

(c) x_{1}, x_{2} ∉ B.

(d) x_{1}, x_{2} ∈ ϕ^{−1}(X^{δ}).

(e) For i = 1, 2, Θ_{e}(ϕ_{i}(x_{1}), ϕ_{i}(x_{2})) ≥ λ_{0} > 0.

(f) d(ϕ(x_{2}), e) ≥ d(ϕ(x_{1}), e) ≥ r with r sufficiently large (depending on K, δ, ϵ, λ_{0}).

Then there exist constants δ′ (depending on K, δ) and Λ (depending on λ_{0}) such that

d(e, F (β, γ)) ≤ Λ, with β, γ ∈ L_{δ′}(ϕ).

Proof. Let δ′ = δ/η_{1} where η_{1} is as in Proposition 4.10. Then by Proposition 4.10 there exist δ′-limit points β_{1}, β_{2} so that for i = 1, 2, j = 1, 2,

Θ_{e}(ϕ_{i}(x_{j}), π_{i}(β_{j})) ≤ e^{−ν_{2}d_{ρ}(x_{j},0)}.

As r increases, so does d_{ρ}(x_{j}, 0) by (3). Then for sufficiently large r, Θ_{e}(ϕ_{i}(x_{j}), π_{i}(β_{j})) ≤ λ_{0}/4, so by the triangle inequality, Θ_{e}(π_{i}(β_{1}), π_{i}(β_{2})) > λ_{0}/2. Then by Lemma 6.1, e must be a bounded distance from the geodesic connecting π_{i}(β_{1}) to π_{i}(β_{2}). Let F (β_{1}, β_{2}) be the flat which is the product of these geodesics for i = 1, 2. It is then clear that e is a bounded distance from the flat F (β_{1}, β_{2}).

Choosing constants. Choose ν smaller than a large negative power of κ (say ν = 1/(32κ^{3}) < 1/2). Choose ϵ, δ so that Lemma 4.8 is satisfied with K = κ, and so that condition (b) of Proposition 6.2 is satisfied both with K = κ and K = κ^{2}/ν. Note that ϵ and δ depend only on κ. Now choose R so large and that condition (f) of Proposition 6.2 is satisfied with r = R/(4κ), with ϵ, δ as chosen, and both with K = κ and with K = κ^{2}/ν. We also require that R > 16κ^{2}C/ϵ. Now let δ̄ denote one half of the smaller of the two constants δ′ produced by the two applications (K = κ and K = κ^{2}/ν) of Proposition 6.2.

Method of proof. We will show the following: for every point x_{1} ∈ R^{2}, ϕ(x_{1}) is within a bounded distance of a flat of the form F (β_{1}, β_{2}) where β_{1}, β_{2} belong to the δ̄-limit set Lδ̄(ϕ), and π_{i}(β_{1}) ≠ π_{i}(β_{2}), i = 1, 2. This is enough since the cardinality of the limit set is bounded by Proposition 4.2 (limit set is finite), and the limit set is independent of the point x_{1} by Lemma 4.5.

Separation into cases. Let x_{1} ∈ R^{2} be an arbitrary point. By Lemma 4.8 there exists a point x_{2} ∈ ϕ^{−1}(X^{δ}(ϕ(x_{1}))) with d(x_{1}, x_{2}) = R. Let z denote the midpoint of the geodesic in X between ϕ(x_{1}) and ϕ(x_{2}). We choose coordinates on X so that the origin e is z. By Lemma 4.3 x_{1}, x_{2} ∈ ϕ^{−1}(X^{δ}(e)), hence condition (d) of Proposition 6.2 holds. Since e is the midpoint of the geodesic connecting ϕ(x_{1}) and ϕ(x_{2}), Θ_{e}(ϕ_{i}(x_{1}), ϕ_{i}(x_{2})) = π. Thus condition (e) of Proposition 6.2 also holds. The conditions (b) and (f) hold by the choice of constants. Hence we need to verify the conditions (a) and (c) of Proposition 6.2. There are two cases to consider:

Case 1: d(ϕ(R^{2}), z) > νR. In this case choose coordinates so that x_{1} = 0 ∈ R^{2}. We claim that the map ϕ satisfies (2) and (3) on all of R^{2} with ρ = κ^{2}R and K = κ^{2}/ν.

The equation (2) holds because ϕ is a κ-quasi-isometry, ρ > R > 4C/ϵ. and K > κ. The lower bound in (3) follows like this: For any u ∈ R^{2},

d(ϕ(u), e) ≥ max(d(ϕ(u), ϕ(0)) − d(ϕ(0), e), νR)

≥ max(\frac{1}{κ}d(u, 0) − κR/2 − 3C/2, νR)

≥ max(\frac{1}{κ}d(u, 0) − 2κR/3, νR)

≥ \frac{ν}{κ^{2}} max\{\frac{κ}{ν}(d(u, 0) − 2κ^{2}R/3), κ^{2}R)\} ≥ \frac{ν}{κ^{2}} max\{d(u, 0), κ^{2}R\}

= \frac{1}{K} d_{ρ}(u, 0)

where the second line follows from d(ϕ(0), e) = \frac{1}{2}d(ϕ(x_{1}), ϕ(x_{2})) ≤ \frac{1}{2}(κR+C), and the second to last line following since: If d(u, 0) ≤ κ^{2}R then it is clear; if d(u, 0) > κ^{2}R then \frac{κ}{ν} · \frac{1}{3}d(u, 0) > d(u, 0) since κ > 3ν.

The upper bound in (3) follows from

d(ϕ(u), e) ≤ d(ϕ(u), ϕ(0)) + d(ϕ(0), e) ≤ κd(u, 0) + \frac{κR}{2} + \frac{3}{2}C

≤ \frac{κ^{2}}{ν} max(d(u, 0), κ^{2}R) = Kd_{ρ}(u, 0) (16)

since ν < 1, κ > 1 and R > 4C/ϵ > 4C.

Thus condition (a) of Proposition 6.2 is verified in case 1. Condition (c) of Proposition 6.2 is vacuous since B is empty.

Case 2: There exists p ∈ ϕ(R^{2}) such that d(p, e) ≤ νR.

Choose coordinates on R^{2} so that ϕ^{−1}(p) is the origin 0. We claim that ϕ satisfies (2) and (3) with K = κ and ρ = 0 outside the ball B = B(0, r′) where r′ = max(2C/ϵ, 2κ(νR + C)). Indeed (2) is satisfied outside B because ϕ is a (κ, C) quasi-isometry, and for points u outside B, ϵd(u, 0) ≥ 2C. The lower bound of (3) is satisfied outside B because if u ∉ B then d(u, 0) ≥ 2κ(νR + C), hence

d(ϕ(u), e) ≥ d(ϕ(u), p) − d(p, e) ≥ \frac{1}{K} d(u, 0) − d(p, e) − C
≥ \frac{1}{K} d(u, 0) − (νR + C) ≥ \frac{1}{2K} d(u, 0)  (17)

and the upper bound following similarly. Thus condition (a) of Proposition 6.2 is satisfied.

Note that for i = 1, 2,

d(x_{i}, ϕ^{−1}(p)) ≥ \frac{1}{κ}d(ϕ(x_{i}), p) − C ≥ \frac{1}{κ}d(ϕ(x_{i}), e) − \frac{1}{κ}d(e, p) − C ≥ [\frac{1}{4κ^{2}} − \frac{2ν}{κ} ] R. (18)

In the last inequality we used d(ϕ(x_{i}), e) = \frac{1}{2}d(ϕ(x_{1}), ϕ(x_{2})) ≥ \frac{1}{κ}R − C. By (18) and the choices of ν and R, we know x_{1}, x_{2} ∉ B, so condition (c) of Proposition 6.2 is also satisfied.

Completing the proof. By Proposition 6.2, d(e, F (β, γ)) ≤ Λ. But then

d(ϕ(x_{1}), F (β, γ)) ≤ d(ϕ(x_{1}), z) + d(z, F (β, γ)) ≤ \frac{1}{2}(κR + C) + Λ.

This proves the theorem.

7 Consequences

7.1 Quasi-flats with holes

Suppose Ω ⊂ R^{2}. Let Ω_{ϵ,R} be the set of all x ∈ Ω such that for any y ∈ R^{n} \ B(x, R) the closed ball B(y, ϵd(x, y)) contains points of Ω, i.e there exists z ∈ Ω with d(z, y) ≤ ϵd(x, y). If Ω_{ϵ,R} is non-empty, we think of Ω as a “quasi-flat with holes”.

As a corollary of the proof of Theorem 1.1 we have the following theorem:

Theorem 7.1 (Quasi-flats with holes). Suppose ϕ is a (κ, C) quasi-isometric embedding from Ω ⊂ R^{2} into X = H^{2} × H^{2}. Then there exist constants F = F (κ) and ϵ_{0} = ϵ_{0}(κ) such that if ϵ < ϵ_{0} then, if Ω_{ϵ,R} is non-empty, ϕ(Ω_{ϵ,R}) lies in the Nneighborhood of F flats in X, where N depends on κ, C, and R.

The version of this theorem for arbitrary higher rank symmetric spaces X ([EF]) is is used in [E] to deduce quasi-isometric rigidity for non uniform lattices in semisimple groups without rank one factors.

Proof. Suppose x ∈ Ω_{ϵ,R}. We build a grid as in §2.2, so that each point y is contained in a cell of size between (ϵ/2)d(x, y), and ϵd(x, y). Then, by the definition of Ω_{ϵ,R}, for every cell C with d(C, x) ≥ R, Ω ∩ C is not empty; we pick a point y_{C} in the intersection.

We can now “connecting the dots” (see §2) to construct a continuous function ϕ_{x} defined on all of R^{2} which agrees with ϕ on every y_{C}, and satisfying for all y ∈ Ω − B(x, R)

d(ϕ(y), ϕ_{x}(y)) ≤ 4κϵd(y, x).  (19)

Choose ϵ_{0} to be 1/(8κ) times the ϵ chosen in the proof of Theorem 1.1. We then repeat the proof of Theorem 1.1, except that immediately replace ϕ by ϕ_{x}. The function ϕ_{x} satisfies (2) and (3) because of (19) combined with the fact that ϕ does. Thus the proof of Theorem 1.1^{2} shows that x is within a bounded distance from one of the flats F (β, γ) where β and γ belong to the limit set Lδ̄(ϕ_{x}). But the this limit set is (up to replacing δ̄ by δ̄/2) independent of the choice of basepoint x ∈ Ω_{ϵ,R} because of (19) and an argument analogous to the proof of Lemma 4.6.

7.2 Concluding Remarks

It is possible to use Theorem 1.1 to prove that any quasi-isometry of X = H^{2} × H^{2} lies a bounded distance from a product of quasi-isometries of H^{2}. The idea is that Theorem 1.1 implies that, under a quasi-isometry ϕ : X → X, the image of a flat lies close to a finite union of flats. Since the domain and range contain many flats, it is possible to make arguments using the geometry of X to show that this finite union actually consists of a single flat. Once one knows that ϕ take flats to flats, the result (in the case X = H^{2} × H^{2}) is an easy exercise.

For more details we refer the reader to [EF], where the argument is worked out for more general X.

References

[E] A. Eskin, Quasi-isometric rigidity of nonuniform higher rank lattices, in preparation.

[EF] A. Eskin and B. Farb, Quasi-flats and rigidity in higher rank symmetric spaces, preprint, October 1995.

[FS] B. Farb and R. Schwartz, The large-scale geometry of Hilbert modular groups, 1994 preprint (submitted).

2To handle case 2 in the proof of Theorem 1.1 we must remark that if x ∈ Ω_{ϵ,R} then so is any point p as long as R is sufficiently large compared to ϵ and d(x, p).