FILLING-INVARIANTS AT INFINITY FOR MANIFOLDS OF NONPOSITIVE CURVATURE

NOEL BRADY AND BENSON FARB

0. Introduction

Homological invariants “at infinity” and (coarse) isoperimetric inequalities are basic tools in the study of large-scale geometry (see e.g., [Gr]). The purpose of this paper is to combine these two ideas to construct a family div_{k}(X^{n}), 0 ≤ k ≤ n − 2 , of geometric invariants for Hadamard manifolds X^{n} ^{1} . The div_{k}(X^{n}) are meant to give a finer measure of the spread of geodesics in X^{n}; in fact the 0-th invariant div_{0}(X^{n}) is the well-known “rate of divergence of geodesics” in the Riemannian manifold X^{n}.

The definition of div_{k}(X^{n}) goes roughly as follows (see Section 1 for the precise definitions): Find the minimum volume of a ball B^{k+1} needed to fill a sphere S^{k}, where S^{k} sits on the sphere S(r) of radius r in X^{n}, and the filling ball B^{k+1} is required to lie outside the open ball B(r)^{∘} in X^{n}. Then div_{k}(X^{n}) measures the growth of this volume as r → ∞; hence div_{k}(X^{n}) is in some sense a k -dimensional isoperimetric function at infinity.

We view the invariants div_{k}(X^{n}) in the same way as we view the standard isoperimetric inequalities (for manifolds or for groups): as basic geometric quantities to be computed.

The div_{k}(X^{n}) are quasi-isometry invariants of X^{n}. The fundamental group π_{1}(M^{n}) (endowed with the word metric) of a compact Riemannian manifold is quasiisometric to the universal cover M̃^{n}; hence the div_{k}(M̃^{n}) give quasi-isometry invariants for fundamental groups of closed, nonpositively curved manifolds M^{n}.

The contents of this paper are as follows: In Section 1, div_{k}(X^{n}) is defined and shown to be a quasi-isometry invariant. The core of this paper (Sections 2,3,4) describes three geometric techniques for computing div_{k}(X^{n}) for some basic examples. Section 4 also explores some surprising quasi-isometric embeddings hyperbolic spaces and solvable Lie groups into products of hyperbolic spaces.

Farb is supported in part by an NSF Postdoctoral Fellowship. His work at MSRI is supported by NSF grant DMS-9022140.

1Recall that a Hadamard manifold is a complete, simply-connected manifold with nonpositive sectional curvatures.

We would like to express our gratitude to K. Fujiwara, C. Pugh, and A. Wilkinson for useful discussions, and to G. Kuperberg for doing the first two figures.

1. Definitions and Quasi-isometry Invariance

Let X^{n} be a Hadamard manifold; that is, a complete, simply connected manifold all of whose sectional curvatures are nonpositive. Let S(r) , B(r) and B(r)^{∘} denote respectively the sphere, ball and open ball of radius r about a fixed basepoint x_{0} of X^{n}, and let C(r) = X^{n} \ B(r)^{∘}. Note that C(r) deformation retracts onto the sphere S(r) ; hence any continuous map f : S^{k} → S(r) admits a continuous extension, or filling f̂: B^{k+1} → C(r) , for any integer 0 ≤ k ≤ n − 2 .

We shall be considering lipschitz maps to the manifold X^{n}. By the Whitney Extension Theorem, we know that if f as above is lipschitz, then the extension f̂ of f can be chosen to be lipschitz (with the same lipschitz constant). By Rademacher’s Theorem, lipschitz maps are differentiable almost everywhere, enabling one to define the k -volume of f : S^{k} → X^{n} and the (k + 1) -volume of f̂: B^{k+1} → X^{n}, where S^{k} and B^{k+1} denote the unit sphere and ball in euclidean space R^{n}. More precisely, if the derivative D_{x}f exists at a point x ∈ S^{k}, it sends an orthonormal basis at x to a k -tuple of vectors in T_{f(x)}(X^{n}) . We can comupte the k -volume of the parallelopiped spanned by this k -tuple using the metric on X^{n}. This defines a function V (x) almost everywhere on S^{k}, and we can then define the k -volume of f , denoted vol_{k}(f) , to be the integral of V over S^{k}. This integral exists because V (x) is a bounded measurable function defined almost everywhere on S^{k}, as ∥D_{x}f∥ is bounded by the lipschitz constant of f .

We are now ready to define the invariant div_{k}(X^{n}) for a fixed integer 0 ≤ k ≤ n − 2 . Although the concept of div_{k} is quite simple, the precise definition of div_{k} needs to be somewhat technical in order to make it manifestly a quasi-isometry invariant. This is accomplished using a variation of a trick introduced in [Ge].

Let A > 0 and 0 < ρ ≤ 1 be given. For r > 0 , we define a map f : S^{k} → S(r) to be A -admissible if:

• f is lipschitz, and

• vol_{k}(f) ≤ Ar^{k}

and say that the extension f̂ of f is ρ -admissible if:

• f̂ is lipschitz, and

• f̂(B^{k+1}) ⊂ C(ρr) . In other words, the only admissible fillings are those which lie outside the open

ball B^{∘}(ρr) in X^{n}. Now define

δ_{ρ,A}^{k}(r) = sup_{f} inf ; _{f} vol_{k+1}(f̂)

FILLING-INVARIANTS AT INFINITY FOR MANIFOLDS OF CURVATURE ≤ 0 3

where the supremun and infimum are taken over A -admissible maps f and ρ - admissible fillings f̂ of f . We call the resulting two-parameter family of functions

div_{k}(X^{n}) = {δ_{ρ,A}^{k} : 0 < ρ ≤ 1, A > 0}

the k -th divergence of X^{n} with respect to the point x_{0} . The parameters ρ and A are necessary in order to make div_{k} into a quasi-isometry invariant (see Theorem 1.1).

Remark: We note that the function δ_{ρ,A}^{k}(r) , as a sup of an inf , may not be realized by an actual filling, though of course there are (admissible) fillings arbitrarily close to realizing this function. We will ignore this distinction in what follows, as we are only interested in the growth of δ_{ρ,A}^{k}(r) .

In this paper we shall only be concerned with distinguishing between polynomial and exponential functions. Hence the following equivalence relation: given functions f, g : R^{+} → R^{+} , we write f ⪯ g if there exist constants a, b, c > 0 and an integer s ≥ 0 such that f(x) ≤ ag(bx) + cx^{s} for all sufficiently large x. Now write f ∼ g if both f ⪯ g and g ⪯ f . This defines an equivalence relation on the class of functions from R^{+} → R^{+} , and it makes sense to call the equivalence classes polynomial, exponential, super-exponential, etc.

Similarly, one defines an equivalence relation among k -th divergences as follows: say that div_{k} ⪯ div′_{k} if there exist 0 < ρ_{0}, ρ′_{0} ≤ 1 and A_{0}, A′_{0} > 0 such that for every pair (ρ, A) with ρ < ρ_{0} and A > A_{0} there exist ρ′ < ρ′_{0} and A′ > A′_{0} with δ_{ρ,A}^{k} ⪯ δ′_{ρ′,A′}^{k} . Now define div_{k} ∼ div′_{k} if we have both div_{k} ⪯ div′_{k} and div′_{k} ⪯ div_{k}. In particular, we say that div_{k} is polynomial or exponential, written div_{k} = poly or div_{k} = exp, if there exists 0 < ρ_{0} ≤ 1 and 0 < A_{0} such that δ_{ρ,A}(r) ∼ r^{d} (for some integer d > 0 ) or δ_{ρ,A}(r) ∼ e^{r} for all ρ < ρ_{0} and A > A_{0} . Thus one can speak of polynomial or exponential k -th divergences.

We are now ready to prove that the invariants div_{k}(X^{n}) are actually quasi-isometry invariants of X^{n}, sometimes called geometric invariants. Recall that a quasi-isometry is basically a coarse bi-lipshitz map; these are the appropriate maps to study when one is interested in large-scale geometric properties of a space, or in geometric properties of the fundamental group of a compact Riemannian manifold (see, e.g., [Gr]). More precisely, we recall the following:

Definition: Let X and Y be metric spaces. A quasi-isometry is a pair of maps f : X ⟶ Y, g : Y ⟶ X such that, for some constants K, ϵ, C > 0 :

\begin{matrix}d_{Y} (f(x_{1}), f(x_{2})) ≤ Kd_{X}(x_{1}, x_{2}) + ϵ, & d_{X}(g ∘ f(x_{1}), x_{1}) ≤ C \\ d_{X}(g(y_{1}), g(y_{2})) ≤ Kd_{Y} (y_{1}, y_{2}) + ϵ, & d_{Y} (f ∘ g(y_{1}), y_{1}) ≤ C \end{matrix}

for all x_{1}, x_{2} ∈ X, y_{1}, y_{2} ∈ Y . Note that neither f nor g need be continuous. If such maps exist, X and Y are said to be quasi-isometric; the map f is called a K - quasi-isometry. A quasi-isometric embedding is defined similarly. A basic example to keep in mind is that the fundamental group π_{1}(M) (endowed with the word metric) of a compact Riemannian manifold M is quasi-isometric to the universal cover M̃ of M .

Theorem 1.1 ( div_{k} is a quasi-isometry invariant). The k -th divergence, div_{k}, is a quasi-isometry invariant of Hadamard manifolds. In particular, div_{k} gives a quasi-isometry invariant for fundamental groups of closed, nonpositively curved manifolds.

Theorem 1.1 allows us to simply speak of the k -th divergence div_{k}(X^{n}) as “polynomial” or “exponential”, denoted by div_{k}(X^{n}) ∼ exp and div_{k}(X^{n}) ∼ poly, respectively, without having to speak of the actual 2-parameter family of functions by which div_{k}(X^{n}) is defined. Whether div_{k}(X^{n}) is polynomial or exponential is then a quasi-isometry invariant notion.

Proof of Theorem 1.1: Let F : (X, x_{0}) → (Y, y_{0}) and G : (Y, y_{0}) → (X, x_{0}) be L -lipschitz maps between two based Hadamard manifolds which are determined by a quasi-isometry between X and Y (see Appendix A for existence of lipschitz quasi-isometries). We shall use these maps to compare the k -th divergences div_{k} for X and div′_{k} for Y .

Given an A -admissible map f : S^{k} → S_{X}(r) , we can compose with F to get a lipschitz map F ∘ f : S^{k} → Y with (F ∘ f)(S^{k}) ⊂ B_{Y} (Lr) ∩ C_{Y} (r/L) . Note that vol_{k}(F ∘ f) ≤ A(Lr)^{k}. Radial projection onto S_{Y} (r/L) defines a volume nonincreasing lipschitz map (lipschitz constant 1) π : C_{Y} (r/L) → S_{Y} (r/L) , and so the composition (π ∘ F ∘ f) : S^{k} → S_{Y} (r/L) is AL^{2k} -admissible (see Figure 1). There is an admissible filling of this map with (k + 1) -volume bounded by δ_{ρ,A/L^{2k}}^{′k}(r/L) .

We obtain a lipschitz extension of F ∘ f as follows: on the radius one-half ball in B^{k+1} take the map ({ \atop π ∘ F ∘ f}) ∘ d_{2} where d_{2} denotes the dilation taking the radius one-half ball \frac{1}{2}B^{k+1} onto B^{k+1} , and on the remaining annular region just interpolate between the maps F ∘ f and ({ \atop π ∘ F ∘ f}) ∘ d_{2} (sending radial geodesics in B^{k+1} to geodesic segments between image points in Y ). Note that the (k + 1) -volume of this map is bounded by δ′_{ρ,AL^{2k}}(r/L) plus a polynomial of degree k + 1 in r. This polynomial bounds the (k + 1) -volume of the annular region, and is the reason we use the equivalence relation among functions defined above.

Now postcomposition with G yields a lipschitz map from B^{k+1} to X which lies outside the ρr/l^{2} -ball about x_{0} , and the restriction of this map to S^{k} is a constant distance (pointwise) away from the original map f : S^{k} → X, as G∘F is a constant distance away from the identity map 1_{X} . It is easy to see that one can interpolate between these maps to obtain a lipschitz map f̂ : B^{k+1} → X which is a ρ/L^{2} - admissible filling of f . Note that

vol_{k+1}(f̂) ≤ L^{k+1}δ′_{ρ,AL^{2k}}(r/L) + p(r)

Figure 1. This figure illustrates part of the proof of Theorem 1.1.

where p(r) is a polynomial of degree k + 1 in r, and so

δ_{ρ/L^{2},A}(r) ⪯ δ′_{ρ,AL^{2k}}(r) .

Thus, div_{k} ⪯ div′_{k}. Similarly, div′_{k} ⪯ div_{k} and so div_{k} is a true quasi-isometry invariant.

□

The proof of Theorem 1.1 shows that div_{k} can be made more precise than poly or exp; in fact, div_{k} is a well-defined quasi-isometry invariant up to an additive factor of x^{k+1} .

2. Suspending Hard-to-Fill Spheres

In this section we show how to suspend hard-to-fill spheres in X to hard-to-fill spheres in X ×R . This provides a lower bound for the (k+1) st-divergence of X ×R in terms of the k th-divergence for X.

Theorem 2.1 (suspending hard-to-fill spheres). Let (X^{n}, x_{0}) be a based Hadamard manifold. Then

div_{k+1}(X^{n} × R) ≥ div_{k}(X^{n}) for any 0 ≤ k ≤ n − 3.

Theorem 2.1 shows, for example, that

div_{2}(H^{2} × R^{2}) ≥ div_{1}(H^{2} × R) ≥ div_{0}(H^{2}) ∼ exp.

Proof of Theorem 2.1: Denote X^{n} simply by X, and let S(r) be the sphere of radius r in X × R . Since X = X × {0} is totally geodesic in X × R , the intersection S′(r) = S(r) ∩ X is the sphere of radius r in X. Choose an admissible map f : S^{k} → S′(r) which realizes div_{k}(X) . We now define a map

Σ(f) : Σ(S^{k}) = S^{k+1} ⟶ S(r)

where Σ(S^{k}) denotes the suspension S^{k} × [0, 1]/((S^{k} × {0}) ∪ (S^{k} × {1})) of S^{k}. Geometrically, define the map Σ(f) as follows: for p ∈ S^{k}, let F_{p} be the 2-flat which is the product of the infinite geodesic in X passing through x_{0} and f(p) and the infinite geodesic {x_{0}} × R . We think of each F_{p} based at the “origin” (x_{0}, 0) ∈ X × R , and note that the points x = (x_{0}, r) and y = (x_{0}, −r) of S(r) are contained in each F_{p}. Let γ_{p} denote the arc of the circle of radius r in F_{p} from x to y; this arc has length πr (see Figure 2).

Figure 2. The arc γ_{p} is used to define Σ(f) at the point {p} × [0, 1] of the suspension Σ(S^{k}) = S^{k+1} .

Define the map Σ(f) via:

Σ(f)({p} × [0, 1]) = γ_{p}.

So, for example, Σ(f) stretches each {p} × [0, 1] by a factor of πr. It is not difficult to check that Σ(f) : S^{k+1} → S(r) is an admissible map. Note also that Σ(f)(S^{k+1}) ∩ (X × {0}) = f(S^{k}) .

Suppose that { \atop Σ(f)} : B^{k+1} → C(r) is an admissible filling of Σ(f) . Then { \atop Σ(f)} ∩ (X × {0}) gives an admissible filling of f(S^{k}) in X, and this filling has volume at least δ^{k}(r) , where δ^{k}(r) is the appropriate divergence function as in the definition of div_{k}(X^{n}) .

Let a constant τ << 1 be fixed. The reasoning above shows that for any 0 ≤ ϵ ≤ τ , the map given by { \atop Σ(f)}∩(X ×\{ϵ\}) gives an admissible filling of f(S^{k}) in X ×\{ϵ\}, and this filling has volume at least δ^{k}(r − ϵ) . Since the leaves {X × {ϵ}}_{0≤ϵ≤τ} are parallel, we have

vol({ \atop Σ(f)}) ≥ ∫_{0}^{τ} δ^{k}(r − ϵ)dϵ
≥ ∫_{0}^{τ} δ^{k}(r − τ)dϵ
≥ τ · δ^{k}(r − τ)
∼ δ^{k}(r)

as desired.

□

3. Pulling-Off Spheres Along Flats

In this section we give a polynomial upper bound for div_{1}(X × R^{2}) for any Hadamard manifold X (Theorem 3.2). In order to do this we need the following, easily believable technical lemma. The main idea in the proof of Theorem 3.2 may be digested indpendently of the proof of this lemma.

Lemma 3.1 (homotoping off a neighborhood of the origin). Let β : S^{1} → R^{2} be a lipschitz map such that the length of β(S^{1}) is at most Ar. Then it is possible to homotope β to β′ : S^{1} → R^{2} of length at most πAr, so that β′(S^{1}) lies outside the open unit ball in R^{2} . The paths β and β′ are homotopic by a lipschitz homotopy of area at most 2πAr.

Proof: Let B^{∘} and ∂B denote respectively the open unit ball and unit circle in R^{2} . The map β is lipschitz and therefore continuous, and so β^{−1}(B^{∘}) is an open subset of S^{1} . There are three cases to consider.

In the first case β^{−1}(B^{∘}) = ∅, and we take β′ = β and the result is trivial.

In the second case β^{−1}(B^{∘}) = S^{1} . Here we take β′ = τv⃗ ∘ β where τv⃗ : R^{2} → R^{2} is just translation by a vector v⃗ of length 2. The homotopy is given by maps τ_{tv⃗} ∘ β where t ∈ [0, 1] . Since the τ_{tv⃗} are isometries of R^{2} in the Euclidean metric, β′ is lipschitz of length Ar, and the homotopy is clearly lipschitz, of area at most 2Ar.

Finally, β^{−1}(S^{1}) may be a non-empty proper open subset of S^{1} , and so is the disjoint union of a collection of open intervals in S^{1} . Given such an open interval (a, b) ⊂ S^{1} , we may define β′|_{[a,b]} to agree with β on the endpoints a and b, and to map [a, b] uniformly over the smaller of the arcs of ∂B determined by β(a) and β(b) (either arc if β(a) and β(b) are antipodal). We take the straight line homotopy tβ′(x) + (1 − t)β(x) in R^{2} , between β and β′. It is clear from the construction that β′ and the homotopy are lipschitz, and that the length of β′ is bounded by πAr, and that the area of the homotopy is at most 2πAr. □

Theorem 3.2 (a polynomial filling). Let X be a Hadamard manifold. Then

div_{1}(X × R^{2})

is polynomial of degree three.

Proof: Let an A -admissible map γ : S^{1} → S(r) be given, where S(r) is the sphere of radius r around a chosen basepoint x_{0} ∈ X. Let π : X × R^{2} → R^{2} be the natural projection.

From Lemma 3.1 applied to β = π ∘γ, it is clear that we may homotope γ slightly so that π ∘ γ(S^{1}) lies outside the open unit ball in R^{2} . An admissible filling of this perturbed γ then gives an admissible filling of γ. Since the area of these two maps differs by at most some constant (not depending on r) times r, we may assume without loss of generality that π ∘ γ(S^{1}) lies outside the open unit ball in R^{2} .

We now give an admissible filling γ̂ : B^{2} → C(r) of γ. The filling is given by the tracks of γ under a sequence of 4 homotopies which homotope γ (outside B(r) ) to a point; this filling is illustrated in Figure 3.

Here is the sequence of homotopies :

(1) Radial projection in the R^{2} factor to the 3r -sphere in R^{2} : (x, v⃗) ↦ (x, t · \frac{v⃗}{||v⃗||}), 1 ≤ t ≤ 3r. Since π ∘ γ lies outside the open unit ball in R^{2} , this radial projection is well-defined, and increases the length of γ by at most a factor of 3r (hence the length of the image of γ under this radial projection is at most 3r · |γ| ≤ 3Ar^{2} ).

(2) Coning-off in the X factor: (x, v⃗) ↦ (σ_{x}(t), v⃗), 0 ≤ t ≤ 1 , where σ_{x} is the (unique) geodesic in X from x to x_{0} .

(3) Pulling to the 5r -sphere in the X factor: (x_{0}, v⃗) ↦ (τ(t), v⃗), 0 ≤ t ≤ 1 , where x_{1} is any (fixed) point lying on the sphere of radius 5r in X, and τ is the unique geodesic from x_{0} to x_{1} .

Figure 3. This figure gives a sequence of 4 homotopies which homotope γ (outside B(r) ) to a point. The homotopy is illustrated by its projections onto the X and R^{2} factors.

(4) Coning-off in the R^{2} factor: (x_{1}, v⃗) ↦ (x_{1}, (1 − t) · v⃗), 0 ≤ t ≤ 1 .

It is easy to check that the images of these homotopies lies outside B(r) , since the metric on X × R^{2} is just the product metric. Piecing together these homotopies gives a map γ′ : S^{1} × [0, 1] → C(r) with the image γ′(S^{1} × {1}) being the point (x_{1},0) ; hence this induces a map on the cone on S^{1} , that is a map γ̂ : B^{2} → C(r) . The map γ̂ is easily seen to be an admissible filling of area at most

(3Ar^{2}) · (3r) + (Ar) · r + (3Ar^{2}) · (5r) + (3Ar^{2}) · (3r) ≤ 35Ar^{3}.

Hence div_{k}(X × R^{2}) is (equivalent to) a polynomial of degree 3 . □

Remark: It is probably true, more generally, that div_{k}(X × R^{m}) is polynomial for k < m. The proof of Theorem 3.2 works verbatim in this case, except for Lemma 3.1 (which we believe to be true, although we have not been able to find a proof).

4. Transverse Flats and Hyperbolic Spaces in Products

It is a surprising (but not difficult to see) fact that there is, for example, a quasiisometrically embedded copy of H^{3} inside H^{2} × H^{2} ^{2}. In the first part of this section, we show that there are quasi-isometric embeddings of hyperbolic spaces and solvable Lie groups in products of hyperbolic spaces, although these embeddings are not quasi-convex. We then apply the first embeddings to find a lower bound for certain div_{k} of products of hyperbolic spaces.

The idea is to exploit the fact that, in a product of hyperbolic spaces X, there is a flat and a nicely embedded hyperbolic space whose dimensions add to dim(X)+1 . The flat is used to find a polynomial-volume sphere; the hyperbolic space is used to show that any admissible filling of this sphere has exponential volume.

Recall that a subset Y of a metric space X is called quasiconvex in X if there is a constant D ≥ 0 so that any geodesic in X between points y_{1}, y_{2} ∈ Y lies in the D -neighborhood of Y .

Proposition 4.1 (q.i. embeddings). There are quasi-isometric embeddings of

H^{(m_{1} + ··· + m_{n}) − k+1}

and of an ((m_{1} + ⋯ + m_{n}) − k + 1) -dimensional solvable Lie group in X = H^{m_{1}} × ⋯ × H^{m_{k}} , where each m_{i} > 1 . These embeddings are not quasiconvex.

Examples: Proposition 4.1 shows that there are quasi-isometrically embedded (but not quasiconvex) copies of H^{3} and of the three-dimensional geometry Sol in H^{2} × H^{2} , and that there are quasi-isometrically embedded copies of H^{4} in H^{2} × H^{2} × H^{2} and of H^{5} in H^{3} × H^{3} .

2We somehow remember that this fact was stated by Gromov somewhere in [Gr], but we’ve been unable to locate the exact reference.

Proof: Let γ(t) = (γ_{1}(t), …, γ_{k}(t)) be an infinite diagonal geodesic in X, parameterized by arc length; so that γ_{i} traces out (at a speed of 1/\sqrt{k} times unit speed) an infinite geodesic g_{i} in H^{m_{i}} . For each 1 ≤ i ≤ k, let S_{i}^{+}(t) (respectively S_{i}^{+}(t) ) denote the horosphere in H^{m_{i}} which is centered at γ_{i}(+∞) (respectively γ_{i}(−∞) ) and which contains the point γ(t) .

Now we define the quasi-isometrically embedded copy of hyperbolic space ^{3}

Y = H^{(m_{1}+···+m_{k})−k+1} = R^{(m_{1}+···+m_{k})−k} × R

in H^{m_{1}} × ⋯ × H^{m_{k}} to be the set of points

{(x_{1}, …, x_{k}) : x_{i} ∈ S_{i}^{+}(t) , t ∈ R}

Note that this set contains the geodesic γ which is the vertical R in R^{(m_{1}+···+m_{k})−k} ×R . Each horosphere S_{i} contributes a copy of R^{m_{i}−1} which is scaled by a factor of e^{−t/\sqrt{k}}. Hence the metric induced on R^{(m_{1}+···+m_{k})−k} × R is given by

dt^{2} + e^{−t/\sqrt{k}}ds^{2}

where ds^{2} denotes the Euclidean metric on R^{(m_{1}+···+m_{k})−k}.

First we show that Y ⊂ X is a quasi-isometric embedding. For any two points p, q ∈ Y we have d_{X}(p, q) ≤ d_{Y} (p, q) . For the rest of the proof we refer to Figure 4. Let p_{i} and q_{i} denote the projections of p and q onto the factors H^{m_{i}} , and let γ_{i} denote the geodesic in H^{m_{i}} between p_{i} and q_{i}. Since p, q ∈ Y , the p_{i} all have the same vertical coordinate t and the q_{i} all have the same vertical coordinate t′.

For each 1 ≤ i ≤ k let

π_{i} : H^{m_{i}} → g_{i} : (x_{1}, …, x_{m_{i}}, t) ↦ t

denote the horospherical projection onto the vertical geodesic g_{i}, and let

τ = sup_{i} length(π_{i}(γ_{i})) .

Denote by a_{i} and b_{i} respectively, the points in H^{m_{i}} which lie vertically above p_{i} and q_{i} on the horospherical level min{t, t′} + τ .

Now consider the path in Y from p to q which consists of the diagonal geodesic segments from p = (p_{i}, …, p_{k}) to (a_{i}, …, a_{k}) and from (b_{1}, …, b_{k}) to q = (q_{1}, …, q_{k}) ,

3When using the coordinates R^{n−1} × R for hyperbolic space H^{n} (where t ∈ R acts on R^{n−1} by (x_{1}, …, x_{n}) ↦ e^{−t}(x_{1}, …, x_{n}) ) we shall refer to the R^{n−1} coordinates as horospherical and the R coordinate as vertical.

Figure 4. The hyperbolic space is quasi-isometrically embedded.

and a geodesic (in the intinsic metric on the product of horospheres S_{1}^{+} × ⋯ × S_{k}^{+} ) from (a_{1}, …, a_{k}) to (b_{1}, …, b_{k}) . Each of the diagonal geodesics have length bounded by \sqrt{k}τ and the other geodesic segment has length with bound 2\sqrt{k}.

Thus the length of this path is less than 2\sqrt{k}τ + 2\sqrt{k} and so we have

d_{Y} (p, q) ≤ 2\sqrt{k}τ + 2\sqrt{k}
≤ 2\sqrt{k}dH^{m_{j}} (p_{j}, q_{j}) + 2\sqrt{k}
≤ 2\sqrt{k}d_{X}(p, q) + 2\sqrt{k}

where H^{m_{j}} is the factor realizing the maximum of the lengths of π_{i}(γ_{i}) .

Now we show that Y is not quasi convex in X. Consider the two points (p_{1}, p_{2}, …, p_{k}) and (q_{1}, p_{2}, …, p_{k}) in Y where p_{1} and q_{1} lie on the same horosphere in H^{m_{1}} . The X geodesic between these points is just the geodesic in H^{m_{1}} from p_{1} to q_{1} (with the other coordinates just constant at the point (p_{2}, …, p_{k}) ). Clearly this does not lie in Y ; in fact, the distance from its midpoint (r_{1}, p_{2}, …, p_{k}) to Y is given by

min_{l_{1}+l_{2}=l} \sqrt{(k − 1)l_{1}^{2} + l_{2}^{2}}

where l is the vertical height of the geodesic between p_{1} and q_{1} in H^{m_{1}} . This can be made arbitrarily large by choosing p_{1} and q_{1} far apart.

Finally, note that there are quasi isometrically embedded copies of solvable Lie groups in X; these are just the horospheres in X. More explicitly, for example, define Z in X to be the set of points

Z = {(x_{1}, …, x_{k}) : x_{1} ∈ S_{1}^{+}(t) , x_{i} ∈ S_{i}^{−}(t) (i = 2, …, k) , t ∈ R}

We leave it to the reader to verify that the induced metric on Z = R^{(m_{1} + ··· + m_{k}) − k}×R is given by

dt^{2} + e^{−t/\sqrt{k}}ds_{1}^{2} + e^{t/\sqrt{k}}ds_{2}^{2}

where ds_{1}^{2} denotes the Euclidean metric on R^{m_{1}−1} and ds_{2}^{2} denotes the Euclidean metric on R^{(m_{2}+···+m_{k})−k+1} , and that Z is quasi isometrically embedded in X. □

The geodesic γ of Proposition 4.1 is contained in the k -flat F = γ_{1} × ⋯ × γ_{k}. This k -flat is foliated by parallel geodesics of the form

γ_{(s_{1},... ,s_{k})}(t) = (γ_{1}(t + s_{1}), …, γ_{k}(t + s_{k}))

where s_{1} + ⋯ + s_{k} = 0 . The corresponding family of hyperbolic spaces

Y_{(s_{1},... ,s_{k})} = {(x_{1}, …, x_{k}) : x_{i} ∈ S_{i}^{+}(t + s_{i}) , t ∈ R}

gives a codimension (k−1) foliation of X, where each Y_{(s_{i},... ,s_{k})} intersects the k -flat γ_{1} × ⋯ × γ_{k} in the geodesic γ_{(s_{1},... ,s_{k})} .

Lemma 4.2. For any points p ∈ Y_{(s_{1},... ,s_{k})} and q ∈ Y_{(s′_{1},... ,s′_{k})} we have

d_{X}(p, q) ≥ \sqrt{(s_{1} − s′_{1})^{2} + ⋯ + (s_{k} − s′_{k})^{2}}

Proof: Perpendicular projection of X onto the k -flat γ_{1} × ⋯ × γ_{k} is a distance nonincreasing map which takes p and q to points on the geodesic lines γ_{(s_{1},... ,s_{k})} and γ_{(s′_{1},... ,s′_{k})} respectively. The X -distance between the image points is the same as the

distance in the k -flat, which is bounded below by \sqrt{(s_{1} − s′_{1})^{2} + ⋯ + (s_{k} − s′_{k})^{2}} . □

Theorem 4.3 (some hard-to-fill spheres). Let X = H^{m_{1}}×⋯×H^{m_{k}} , each m_{i} > 1 , be a product of k hyperbolic spaces. Then div_{k−1}(X) = exp.

Proof: Consider the family of hyperbolic spaces {Y_{s}} as above, where s ∈ B(1) , the ball of radius 1 in F . Recall that F = γ_{1} × ⋯ × γ_{k} is a totally geodesic, isometrically embedded copy of R^{k} in X. Also recall that the volume of a sphere of radius r in R^{k} is ar^{k} for some constant a depending only on k.

Now let A > a and r > 0 be given, and let T = F ∩ S(r) . Then T is a (k − 1) - dimensional sphere lying on the sphere S(r) of radius r about the origin in X. Since T ⊂ F and F is a flat in X, we have vol_{k−1}(T) ≤ ar^{k} ≤ Ar^{k}. We claim that any filling of T outside of S(r) has (k + 1) -volume on the order of e^{r}.

To prove this claim, suppose that T̂ is an admissible filling of T . First note that the intersection T ∩ Y_{s} in X has dimension 0; in fact, T ∩ Y_{s} consists precisely of the two points T ∩ γ_{s} = {x_{s}, y_{s}}. Since x_{s} and y_{s} are each within a distance of 1 from antipodal points of S(R) lying on the geodesic γ_{0} , it follows that x_{s} and y_{s} are each within a distance of 1 from antipodal points on the sphere S′(r) of radius r in Y . Now T̂ ∩ Y_{s} is a one-dimensional arc in Y_{s} which connects x_{s} to y_{s} outside of S′(r) . Since Y_{s} is a hyperbolic space, div_{0}(Y_{s}) =exp, so that the arc T̂ ∩ Y_{s} has length at least Ce^{r} for some constant C which is independent of r. We note that since s ∈ B(1) , the constant C may be chosen to work for all Y_{s}.

Now

vol_{k+1}(T̂) = ∫_{B(1)} vol_{k+1}(T̂ ∩ Y_{s})dμ(s)
≥ ∫_{B(1)} Ce^{r}dμ(s)  since |T̂ ∩ Y_{s}| ≥ Ce^{r}
≥ Ce^{r}μ(B(1))  by Lemma 4.2

and we are done.

□

Remark: The reason it is necessary to use Lemma 4.2 is that it is possible to have, for example, a disc foliated by an interval’s worth of lines of length e^{r}, but with the area of the disc being constant. For example consider a long, thing quadrilateral in the hyperbolic plane: it is foliated by lines of length e^{r} for r large, but it’s area is bounded by a universal constant; the reason is that the leaves of the foliation are bent so that they come very close together, on the order of e^{−r}, in fact.

5. Questions

As stated above, we view the invariants div_{k}(X) as basic geometric quantities to be computed. We believe that the invariants div_{k}(X^{n}) are computable for many more examples than are covered in this paper.

Question: Compute div_{k}(X) for symmetric spaces X of noncompact type. The simplest case not covered in this paper is div_{2}(H^{2} ×H^{2}) . We believe that div_{1}(X) ∼ e^{r} for the symmetric space X = SL_{n}(R)/SO_{n}(R) .

Question: Can the invariants div_{k}(X) be used to detect the rank of a (globally) symmetric space X of noncompact type?

Appendix A

In this appendix we give a technical proposition which was needed to show that the div_{k}(X^{n}) are quasi-isometry invariants.

Proposition A.1: Suppose f : X → Y is a quasi-isometry between Hadamard manifolds, and suppose that X admits a cocompact lattice. Then f is a bounded distance from a (continuous) lipshitz map f′ : X → Y ; that is, sup_{x∈X}d_{Y} (f(x), f′(x)) ≤ C for some constant C > 0 .

Proof: Since X admits some compact quotient M = X/Γ , it is possible to lift a triangulation of M to to a Γ -equivariant triangulation of X. Note that there are finitely isometry types of simplices in this triangulation of X.

The map f′ is defined inductively on the skeleta of the triangulation. On vertices we simply define f′ to equal f . Suppose f′ is defined on the k -skeleton of the triangulation; then for each (k + 1) -simplex σ, we have a Lipshitz map defined on ∂(σ) , which is a sphere. This map extends to a lipshitz map on σ by Whitney’s Extension Theorem. Do this for each different (k + 1) -simplex σ; the point is that there are only finitely many different lipshitz constants since there are only finitely many isometry types of simplices; hence the map f′ is lipshitz with constant the maximum of the Whitney lipshitz constants on the finitely many isometry types of simplices. □

References

[Ge] S.M. Gersten, Quadratic divergence of geodesics in CAT (0) spaces, Geometric and Functional Analysis, Vol.3, No.6 (1993).

[Gr] M. Gromov, Asymptotic Invariants of Infinite Groups, in Geometric Group Theory, G. Niblo and M. Roller, eds., LMS Lecture Notes 182, Cambridge Univ. Press, 1993.

Noel Brady, Math Department, University of Utah, Salt Lake City, Utah 84112 E-mail address: brady@math.utah.edu

Benson Farb, Math Department, University of Chicago, 5734 University Ave., Chicago, Illinois 60637

E-mail address: farb@math.uchicago.edu