On the Hausdorff dimension of CAT($\kappa$) surfaces

We prove that a closed surface with a CAT($\kappa$) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally, we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(-1) manifolds.


Introduction
Let Σ be a closed surface, and let d be a locally CAT(κ) metric on Σ. One quantity of natural interest is the Hausdorff dimension of (Σ, d), denoted dim H (Σ, d). This dimension is bounded below by 2, the topological (covering) dimension of Σ. However, for an arbitrary metric on Σ there is no upper bound; this can be seen by 'snowflaking' the metric -replacing d(x, y) with d ′ (x, y) = d(x, y) α for 0 < α < 1, which raises the the dimension by a factor of 1 α (see, e.g. [TW05]). In this paper we examine the restriction placed on dim H (Σ, d) by the CAT(κ) condition, and prove the following theorem: Theorem 1. Let (Σ, d) be a CAT(κ) closed surface. Then dim H (Σ, d) = 2. Moreover, there exists some δ 0 > 0 such that for all 0 < δ ≤ δ 0 , inf p∈Σ H 2 (B(p, δ)) > 0 and sup p∈Σ H 2 (B(p, δ)) < ∞ where H 2 denotes the 2-dimensional Hausdorff measure and B(p, δ) is the ball of radius δ around p.
We note that the second statement of the theorem implies the first, but not vice versa. Indeed there are metric spaces with Hausdorff dimension d whose ddimensional Hausdorff measures are zero or infinite.
We became interested in this question for Hausdorff measures, in particular the uniform bounds on the measures of balls, while thinking about some results on entropy for geodesic flows on CAT(−1) manifolds. We include a discussion of these results in the final section of the paper, but here we state the two main theorems we prove.
Theorem 2. Let (Σ, d) be a closed surface with a CAT(0) metric. Let φ t be the geodesic flow on the space of geodesics for (Σ, d). Then the topological entropy for the flow equals the volume growth entropy for the Hausdorff 2-measure.
Theorem 3. Let (X, d) be a closed CAT(−1) manifold (not necessarily Riemannian), and suppose that X admits a Riemannian metric g so that (X, g) is a locally symmetric space. Let h top (φ d t ) and h top (φ g t ) be the topological entropies for the geodesic flows under the two metrics. Then h top (φ d t ) ≥ h top (φ g t ) and if equality holds, (X, d) is also locally symmetric. If dim X > 2, (X, d) and (X, g) are isometric.
Theorem 2 is a version of Manning's [Man79] analogous result for Riemannian manifolds of non-positive curvature, and relies on some work of Leuzinger [Leu06]. Theorem 3 follows from our Theorem 2 and a rigidity result of Bourdon ([Bou96]). Our main work is to note how, via Theorem 2, Bourdon's theorem can be recast as a topological entropy rigidity theorem. This fact may well be known to experts, but we have not found it addressed in the literature.
The paper is organized as follows. In Sections 2 and 3 we show that small distance spheres around each point in Σ are topological circles, and that they are rectifiable with bounded length. In Section 4 we prove Theorem 1, and in Section 5 we discuss the extension of Main Theorem to higher dimensions, and give an example which indicates some of the complications in doing so. In Section 6 we give the proof of topological entropy rigidity (Theorem 3) for CAT(−1) manifolds.
Acknowledgements. We would like to thank Enrico Leuzinger and Mike Davis for helpful conversations. The first author would like to thank Ohio State University for hosting him during the time that most of this work was done. The second author was partially supported by the NSF, under grant DMS-1510640.

The topology of small distance spheres
Let S(p, ǫ) = {z ∈ Σ ∶ d(p, z) = ǫ} and B(p, ǫ) = {z ∈ Σ ∶ d(p, z) ≤ ǫ} respectively denote the metric ǫ-sphere and ǫ-ball centered at p. In this section we prove for small ǫ, all S(p, ǫ) are topological circles. We note that the argument only works for surfaces. In Section 5 we give examples of higher-dimensional CAT(−1) manifolds where the analogous statement is not true.
Throughout this section, we work at small scale. We fix ǫ 0 > 0 small enough so that the following two conditions are satisfied: • ǫ 0 ≤ D κ 2 where D κ is the diameter of the model space of constant curvature κ, and • For all p ∈ Σ, B(p, ǫ 0 ) is (globally) CAT(κ). At these scales, B(p, ǫ 0 ) is locally uniquely geodesic -in particular there is a unique geodesic from p to any point in B(p, ǫ 0 ), which varies continuously with respect to the endpoints. This will be a key fact in the work below. As a consequence, each such ball B(p, ǫ 0 ) is contractible, hence lifts isometrically to the universal cover (Σ,d).
The following Lemma will be useful. Its proof, which is straightforward and can be adapted to any dimension, can be found in [BH99, Proposition II.5.12].
Lemma 4. Let [xy] be a geodesic segment in Σ connecting an arbitrary pair of points x and y. Then [xy] can be extended beyond y. That is, there is a geodesic segment (not necessarily unique) [xy ′ ] properly containing [xy] as its initial segment.
Remark. Using the compactness of X, and a connectedness argument on R, this lemma implies that each geodesic segment [xy] can be infinitely extended.
The main result of this section is the following: Proposition 5. Let Σ be a complete CAT(κ) surface. Then for all ǫ < ǫ 0 , S(p, ǫ) is homeomorphic to the circle S 1 .
In order to establish this result, we use a well-known characterization of the circle S 1 . The circle is the only compact, connected, metric space (X, d) with the property that for any pair of distinct points a, b ∈ X, the complement X ∖ {a, b} is disconnected (see, e.g [HY88, Theorem 2-28]). Let p ∈ Σ be an arbitrary point in Σ, and to simplify notation, we set S ǫ ∶= S(p, ǫ). We now claim that for ǫ < ǫ 0 , S ǫ is homeomorphic to a circle.
Proof. S ǫ is a closed subset of the compact metric space Σ, so it is compact and metric. Since ǫ < ǫ 0 , S ǫ lifts homeomorphically to a subset ofΣ. Since Σ is a surface,Σ is homeomorphic to R 2 or S 2 , so we may take S ǫ to a be a compact subset of R 2 or S 2 .
S ǫ has diameter < D κ , so we may find a path S in R 2 or S 2 homeomorphic to S 1 bounding a disk containing S ǫ and remaining in B(p, ǫ 0 ). Let proj ∶ S → S ǫ be the nearest point projection (for the CAT(κ) metric lifted from Σ). This is a continuous map. Since geodesics inΣ are infinitely extendible, for any point z on S ǫ , the geodesic segment [pz] extends to a geodesic which hits S at a point q. Then proj(q) = z and so this map is also surjective. The surjective, continuous map from the path-connected set S to S ǫ proves that the latter is path-connected.
Proof. As in the proof of Lemma 6, let S in R 2 or S 2 be a Jordan curve containing S ǫ and remaining in B(p, ǫ 0 ). In particular, from the Schoenflies theorem, we can now view S ǫ as contained inside a closed topological disk D 2 (the curve S along with its interior). Given the two distinct points a, b ∈ S ǫ , extend the two geodesic segments Note that [pa ′ ], [pb ′ ] might coincide on some subgeodesic originating from p. Let w be the point at which these two geodesic segments separate, and consider the concatenation [a ′ w] with [wb ′ ]. By uniqueness of geodesics in B(p, ǫ 0 ), we have that [a ′ w] ∩ [wb ′ ] = {w}, and hence they concatenate to give an embedded arc joining the pair of points a ′ , b ′ ∈ S = ∂D 2 . Moreover, the interior (a ′ b ′ ) of the arc is contained in the interior of D 2 . In follows that (a ′ b ′ ) separates the interior of the disk into two connected components U 1 , U 2 . (See Figure 1.) Now by way of contradiction, let us assume S ǫ ∖{a, b} is connected. Then without loss of generality, S ǫ ∩ U 1 must be empty. On the other hand, U 1 is homeomorphic to an open disk, whose boundary is a Jordan curve (formed by the arc (a ′ b ′ ) in the interior of D 2 , along with the portion of the boundary S joining a ′ to b ′ ). The boundary of U 1 contains the arc (wa ′ ) passing through a, and the distance to p varies continuously along (wa ′ ) from a number < ǫ (since w ≠ a) to a number > ǫ (since a ≠ a ′ ). Pick an arc η inside U 1 joining w to a ′ , and consider the distance function restricted to η. It varies continuously from < ǫ to > ǫ, but since S ǫ ∩U 1 = ∅, is never equal to ǫ. This is a contradiction, completing the proof.
Using the topological characterization of S 1 , Proposition 5 now follows immediately from Lemma 6 and Lemma 7.

The geometry of small distance spheres
We now want to prove that for all p ∈ Σ and ǫ < ǫ 0 , S ǫ is rectifiable, and that the lengths of these circles can be uniformly bounded above.
Note that if ǫ ′ < ǫ, the rectifiability of S ǫ implies that S ǫ ′ is also rectifiable. This follows from the fact that the nearest-point projection π Z to a complete, convex subset Z is distance non-increasing a ball of radius < ǫ 0 in a CAT(κ) space (see, e.g. [BH99, Prop. II.2.4 (or the exercise following for κ > 0)]). Applying this to the complete convex subset Z ∶= B(p, ǫ ′ ), and using the (global) CAT(κ) geometry in B(p, ǫ 0 ), we see that π Z is just the radial projection towards p. In particular the image of π Z lies on S ǫ ′ . Next, fix any geodesic γ through p and denote by N (n) and S(n) its two intersections with S ǫ n (chosen so that all N (n) lie on the same component of γ ∖ {p}). Since S ǫ n is a circle, the pair {N (n), S(n)} divides S ǫ n into two arcs, whose closures we call arc E (n) and arc W (n); choose these so that the relative positions of N (n), S(n), arc E (n) and arc W (n) correspond to the cardinal directions on a compass.
Note that for all n, at least one of l(arc E (n)), l(arc W (n)) must be infinite, since l(S n ) is infinite. Without loss of generality, we may assume that l(arc E (n)) = ∞ for infinitely many n, and hence (by the remark above) for all n. We focus our attention now on the family {arc E (n)} n∈N .
On arc E (1) define the following equivalence relation: we declare x ∼ y if there exists some n such that the arc in arc E (n) with endpoints [px] ∩ arc E (n) and [py] ∩ arc E (n) is of finite length. That this is an equivalence relation is easy to check. We denote equivalence classes by [x].
We note two things about arc E (1) ∼ and its equivalence classes. First, N (1) ≁ S(1), for otherwise arc E (n) would have finite length for some n. Second, for each x ∈ arc E (1), [x] is an interval (possibly degenerate). This is because geodesics are unique at the scale we work at, and if three points are arranged around arc E (1) in Thus the decomposition of arc E (1) into the equivalence classes of ∼ is a decomposition into at least two disjoint subintervals (possibly degenerate) of the half-circle arc E (1).
By connectedness of arc E (1), either [N (1)] or [S(1)] is a singleton, or some equivalence class has a closed endpoint in the interior of arc E (1). Let z * be this endpoint or the singleton N (1) or S(1).
If z * is an endpoint of arc E (1), let q be any other point in arc E (1). If z * is the closed endpoint of [z * ] in the interior of arc E (1), let q be any point in arc E (1) which lies on the z * -side of [z * ]. We note that there are infinitely many such q, and by the choice of z * and the topology of the half-circle arc E (1), we may take a sequence of such q approaching z * . Observe that, since q and z * are not equivalent, the geodesic segments [pz * ] and [pq] only agree at the point p.
Consider the geodesic segment [z * q]. By the properties of geodesics in the (globally) CAT(κ) set B(p, ǫ), this geodesic segment lies inside B(p, ǫ) and does not cross the geodesic γ which divides the West and East parts of B(p, ǫ). Suppose that [z * q] does not intersect arc E (n) for some n (as in Figure 2). Then the radial Again by the distance non-increasing properties of the projection, since [z * q] has finite length, this would imply z * ∼ q, which contradicts the choice of q. Therefore the geodesic segment [z * q] must intersect arc E (n) for all n. It must therefore hit p, and by uniqueness of geodesics we conclude that [z Now consider B(z * , ǫ). The work above shows that no q chosen as previously described lies in B(p * , ǫ). But this contradicts our observation above that, using the half-circle topology of arc E (1), we may take such q approaching p * . This contradiction concludes the proof.
Let us denote by l(γ) the length of a rectifiable curve γ. Using Lemma 8 and the compactness of Σ we have: Lemma 9. There exist δ 0 > 0 and some uniform C > 0 such that for all z ∈ Σ, l(S(z, δ 0 )) < C.
Proof. Suppose there is no uniform bound on l(S(z, δ 0 2)). Then we may take a sequence of points p n in Σ with l(S(p n , δ 0 2)) ≥ n. Let p * be any subsequential limit point of (p n ) and note that B(p * , δ 0 ) properly contains S(p n , δ 0 2) for sufficiently large n. The unbounded lengths of the latter, plus again the distance non-increasing properties of nearest-point projection, would imply that the length of S(p * , δ 0 ) is infinite, contradicting Lemma 8. This proves the Lemma.

Proof of Theorem 1
We define a particular non-expanding map from B(p, ǫ 0 ) to the ball of radius ǫ 0 in the model space H 2 . This will be a key tool in our proof of Theorem 1.
Define an equivalence relation on the set of geodesic segments starting at p by declaring γ 1 ∼ γ 2 if the Alexandrov angle between these segments at p is 0. Lemma 12. For any p ∈Σ, whereΣ is a CAT(κ) surface, S p (Σ) ≅ S 1 .
Proof. The natural projection from S ǫ to S p (Σ) is continuous and, by Lemma 4, surjective. The fiber over any point in S p (Σ) is easily seen to be a closed interval. Thus S p (Σ) is homeomorphic to a quotient of S 1 , where each equivalence class is a closed interval in S 1 . It is a well-known result that such a quotient space is automatically homeomorphic to S 1 (this can be easily shown using the topological characterization of S 1 used in the proof of Proposition 5). This establishes the Lemma.
We now construct the non-expanding map to the model surface M κ of constant curvature κ. We closely follow the proof of a similar result presented in [BBI01, Proposition 10.6.10], but for the opposite type of curvature bound (curvature bounded below, rather than above).
Proposition 14. Let Σ be a CAT(κ) surface and p any point in Σ. Let ǫ 0 be as above. Then there is a map Proof. By the choice of ǫ 0 , we can work in Σ or lift B(p, ǫ 0 ) homeomorphically tõ Σ. By corollary 13, S p (Σ) is CAT(1). It is homeomorphic to S 1 , so it is easy to see that there is a surjective map g ∶ S p (Σ) → S 1 which is non-expanding: Let K κ p (Σ) denote the κ-cone over S p (Σ) (see, e.g. [BBI01, §10.2.1]). By its definition, and the fact that S p (Σ) is CAT(1), K κ p (Σ) is CAT(κ) ([BBI01, Theorem 4.7.1]). On the ball B(p, ǫ 0 ) define a logarithm map defined as follows: where v is the direction in S p (Σ) of the geodesic segment [px]. From the nonexpanding property of g and the definition of K κ p (Σ), y) for all x, y ∈ B(p, ǫ 0 ). By its definition, log p preserves distance from the origin, and by its definition, log p maps B(p, ǫ) surjectively onto B K −1 p (Σ) (o, ǫ). Again, using the definition of K κ p (Σ), the non-expanding map g ∶ S p (Σ) → S 1 extends to a map G ∶ K κ p (Σ) → M κ , obtained by realizing M κ as the κ-cone over S 1 . The map is non-expanding since g is, and preserves distance from the origin. G sends B K κ p (Σ) (o, ǫ) surjectively to B Mκ (G(o), ǫ) because g is surjective. Then f = G ○ log p is the desired map.
We are now ready to prove Theorem 1.
Let f be the non-expanding map provided by Proposition 14. Since f preserves radial distance from the origin, f (B(p, δ)) ⊆ B(f (p), δ). Fix any ρ > 0 and suppose {U i } is a countable cover of B(p, δ) with diam(U i ) < ρ. Then the collection (p), δ)). The right-hand quantity approaches H 2 (B(f (p), δ)) as ρ → 0. But this is just the volume of a δ ball in M κ , the model space of curvature κ. Since this is independent of the choice of basepoint p, we obtain the desired uniform lower bound on H 2 (B(p, δ)). Now we bound H 2 (B(p, δ)) above. This portion of the proof uses the bound on the length of S δ obtained in Lemma 9. It is sufficient to bound H 2 (B(p, δ 0 )) uniformly above.
Fix ρ < δ 0 . Let E ρ be any finite ρ 2 -spanning subset of S δ0 . The circumference bound allows us to uniformly bound #E ρ . Index x j ∈ E ρ in order around S δ0 . Let T j be the geodesic triangle with vertices p, x j , x j+1 . T j has edges of length δ 0 , δ 0 and < ρ. Let τ j be the corresponding comparison triangle in M κ (see Figure 3). In M κ , let r j be the δ 0 -length edge fromp tox j . Let r(ρ) be the number of ρ-balls centered at points on r j necessary to cover τ j with centers an ρ-spanning set in r j . Let their centers beȳ 1 , . . .ȳ r(ρ) . Note that we can take r(ρ) = C ′ δ0 ρ for C ′ a constant independent of ρ. Now, pickz i on the other δ 0 -length side of τ j and in B(ȳ i , ρ) ∩ B(ȳ i+1 , ρ). Draw in τ j and in T j the zig-zagging segments connecting y i to z i to y i+1 to z i+1 etc. These partition τ j and T j into a union of triangles. In T j each has all three sides of length < ρ, by the choice ofȳ i andz i . By CAT(κ), the corresponding triangles in T j also have all sides of length < ρ, and then, again by CAT(κ), we see that the ρ-balls centered at y i in T j cover T j .

Local geometry of CAT(−1) spaces in higher dimensions
In this section we make a few remarks on the obstructions to extending the local geometry results we proved for surfaces to higher dimensions. We note that the proofs in the previous section rely heavily on the 2-dimensionality of Σ. We do not know if an analogue of Theorem 1 holds for CAT(κ) metrics on closed higher dimensional manifolds. One of the first steps in our proof was Proposition 5, which showed that the small enough metric spheres inside locally CAT(κ) surfaces were homeomorphic to the circle S 1 . The analogous statement fails in dimensions ≥ 5, as the well-known example below shows.
Proposition 15. For each dimension n ≥ 5, there exists a closed n-manifold M equipped with a piecewise hyperbolic, locally CAT(−1) metric, and a point p ∈ M with the property that for all small enough ǫ, the ǫ-sphere S ǫ centered at p is not homeomorphic to S n−1 . In fact, S ǫ is not even a manifold.
Proof. Such examples can be found in the work of Davis and Januszkiewicz [DJ91, Theorem 5b.1]. We briefly summarize the construction for the convenience of the reader. Start with a closed smooth homology sphere N n−2 which is not homeomorphic to S n−2 . Such manifolds exist for all n ≥ 5, and are quotients of S n−2 by a suitable perfect group π 1 (N n−2 ). Take a smooth triangulation of N n−2 , and consider the induced triangulation T on the double suspension Σ 2 (N n−2 ). By work of Cannon and Edwards, Σ 2 (N n−2 ) is homeomorphic to S n . The triangulation T on S n is not a PL-triangulation, as there exists a 4-cycle in the 1-skeleton of the triangulation whose link is homeomorphic to N n−2 . Now apply the strict hyperbolization procedure of Charney and Davis [CD95] to the triangulated manifold (S n , T ).
This outputs a piecewise-hyperbolic, locally CAT(-1) space M . A key point of the hyperbolization procedure is that it preserves the local structure. Since the input (S n , T ) is a closed n-manifold, the output M is also a closed n-manifold. The 4-cycle in T whose link was homeomorphic to N n−2 now produces a closed geodesic γ in M , whose link is still homeomorphic to N n−2 (i.e. the "unit normal" to γ forms a copy of N n−2 ). It follows from this that, picking the point p on γ, all small ǫ-spheres S ǫ are homeomorphic to the suspension ΣN n−2 . Since N n−2 was not the standard sphere, S ǫ ≅ ΣN n−2 fails to be a manifold at the suspension point x, as every small punctured neighborhood of x will have non-trivial π 1 . We refer the reader to [DJ91, Section 5] for more details.
This cautionary example suggests that small metric spheres in high-dimensional locally CAT(κ) manifolds could exhibit pathologies. In view of these results, and the interest in obtaining higher dimensional analogs, we raise the following question.
Question . Let M be a closed n-manifold equipped with a locally CAT(−1) metric of Hausdorff dimension d. Can d ever be strictly larger than n? Do the uniform bound conditions of Theorem 1 hold in higher dimensions?
The authors suspect that examples with d > n do indeed exist in higher dimensions.

Entropy rigidity in CAT(-1)
In this section we present an entropy rigidity result for closed CAT(−1) manifolds. This result generalizes Hamenstädt's entropy rigidity result from [Ham90] to the CAT (−1) setting. It is very closely related to, and in fact relies on, a rigidity result of Bourdon. The main addition to Bourdon's theorem is the connection to topological entropy via a theorem of Leuzinger (generalizing work of Manning). The theorem draws heavily on the work of others, but we have not seen it presented in this form in the literature.
We remark that our work on Hausdorff 2-measure for surfaces, presented in the earlier sections of this paper, was inspired in part by condition (⋆) that features in Leuzinger's theorem below (Theorem 21).
Recall that a negatively curved locally symmetric space has universal cover isometric to H m K , where m ≥ 2 and K is R, C, the quaternions H, or the octonions O (with m = 2). We suppose that the metrics on these spaces are scaled so that the maximum sectional curvature is −1. The topological entropy for the geodesic flow on a compact locally symmetric space modeled on H m K is km + k − 2 where k = dim R K. More generally, if one has a locally CAT(−1) metric d on the manifold M , there is an associated space G(M, d) of geodesics in M : this is the space of locally isometric maps R → M . Via lifting to the universal cover, this space is topologically a quotient of S n−1 × S n−1 × R by a suitable Γ ∶= π 1 (M ) action. There is a natural flow φ d t on G(M, d), given by precomposition with an R-translation. This is called the geodesic flow associated to the metric d. One can again measure the topological entropy of this flow.
The main result of this section is the following restatement of our Theorem 3: Theorem 16. Let (X, d) be a closed n-dimensional manifold equipped with a locally CAT(−1) metric d (not necessarily Riemannian). Suppose that X also supports a locally symmetric Riemannian metric g, under which (X, g) is locally modeled on H m K , normalized to have maximum sectional curvature −1. Then h top (φ d t ) ≥ h top (φ g t ) = km + k − 2 and if equality holds in the above, (X, d) is also locally symmetric. If n > 2, (X, d) and (X, g) are isometric.
This should be compared with the main theorem of [Ham90], which establishes this same result when the metric d is a Riemannian metric with sectional curvature ≤ −1. The key element in this proof is the following theorem of Bourdon, which he notes is a generalization of Hamenstädt's work.
Theorem 17. [Bou95, Théorème 0.3 and following remarks] LetX be a CAT(−1) space with a cocompact isometric action by Γ which also acts convex cocompactly by isometries on a negatively curved symmetric space S = H m K . Let Λ be the limit set in ∂ ∞X of Γ, and let d * and g * denote the visual metrics on ∂ ∞X and ∂ ∞ S, respectively. Suppose that dim H (Λ, d * ) = dim H (∂ ∞ S, g * ). Then there exists an isometric embedding F from S intoX such that ∂ ∞ F (∂ ∞ S) = Λ. If dim R S > 2, then this embedding is Γ-equivariant.
This theorem proves rigidity for the extremal value of dim H (Λ, d * ) -an extremal value which had already been computed by Pansu: Theorem 18. [Pan89, Théorème 5.5] With notation as in Theorem 17, A definition for the visual metrics referenced above can be found in [BH99, §III.H.3]. Here we give a short definition of a distance function which is Lipschitz equivalent to any visual metric. Since Hausdorff dimension can be calculated using any distance function in the Lipschitz equivalence class, this suffices for our purposes.
Definition 19. Let (X,d) be a CAT(−1) metric space and let ζ, η ∈ ∂ ∞X . Fix some basepoint x ∈X and define denotes the bi-infinite geodesic inX with endpoints at ζ and η. It is straightforward to see that the Lipschitz class of d * x is independent of x. We also need the following result of Manning, as generalized to the CAT(−1) setting by Leuzinger.
Definition 20. Let (X, d) be a closed manifold with some metric d and endowed with a measure vol. Let (X,d) denote its universal cover. Then the volume growth entropy of (X,d) with respect to vol is where B(p, R) is the ball of radius R about a point p inX.
Manning shows that this is independent of the choice of p and that, for Riemannian manifolds, the lim sup is in fact a limit.
In the surface case, the validity of condition (⋆) for 2-dimensional Hausdorff measure is established in our Theorem 1, so applying Theorem 21 immediately yields Theorem 2.
Finally, we note the following.
Proposition 22. Let (X, d) be compact and locally CAT(−1), and suppose that vol is a measure on X giving finite, nonzero measure to X. Then where dim H denotes Hausdorff dimension. We now prove Theorem 16.
Proof of Theorem 16. Since X is a manifold, we may fix some Riemannian metric on X, for example the locally symmetric metric g (though any metric will do). This defines a Riemannian volume form vol on X. We first claim that vol satisfies condition (⋆) from Theorem 21. Indeed, take δ 0 so small that any δ 0 -ball for the d-metric on X lifts isometrically to (X,d). Then the volume of such a ball is clearly bounded above by the Riemannian volume of X, which is finite as X is compact. This establishes the uniform upper bound on vol(B d (z, δ)).
Suppose that there is no uniform positive lower bound. Then for any fixed δ > 0, there exists a sequence of points z n in X such that vol(B d (z n , δ)) → 0. By compactness we may extract a convergent subsequence z ni with limit z * . For sufficiently large i, B d (z * , δ 2) ⊂ B d (z ni , δ) and hence vol(B d (z * , δ 2)) = 0. Since d and the Riemannian metric g induce the same topology on X, there exists some ǫ > 0 such that B g (z * , ǫ) ⊂ B d (z * , δ 2). But the Riemannian volume of B g (z * , ǫ) must be strictly positive, giving a contradiction. Therefore, condition (⋆) holds for vol.
Let (S, g) be the negatively curved symmetric space on which (X, g) is locally modelled. Using Theorem 21, Proposition 22 and Theorem 18 and the fact that Γ acts cocompactly on X (and so Λ = ∂ ∞X ), we have Now suppose that h top (φ d t ) achieves the lower bound. Then by Theorem 17, there exists an isometric embedding F from S into X with ∂ ∞ F (∂ ∞ S) = Λ = ∂ ∞X ; if n > 2, this embedding can be taken to be Γ-equivariant. Since X is a manifold with dim(X) = dim(S) and ∂ ∞ F (∂ ∞ S) is the full boundary at infinity ofX, it follows that this isometric embedding is in fact surjective. This finishes the proof.