Coarse CAT(0) Property

• Coarse CAT(0) property is a geometric condition on length spaces that generalizes nonpositively curved and hyperbolic spaces by allowing additive errors in triangle comparisons.
• It provides a unified framework that bridges CAT(0) spaces and Gromov hyperbolic spaces, encompassing non-geodesic and non-proper settings.
• The associated bouquet boundary theory extends classical ideal and Gromov boundaries, ensuring useful topological properties like Hausdorffness and compactness in proper spaces.

The coarse CAT(0) property, also known as the rough CAT(0) (rCAT(0)) property, defines a class of metric spaces that generalizes both CAT(0) (nonpositively curved) and Gromov hyperbolic spaces. Developed to capture the essential features of nonpositive curvature in a coarse geometric setting, the rCAT(0) property introduces relaxed comparison conditions parameterized by a constant $C \geq 0$, allowing for additive errors in the triangle comparison axioms. This framework is robust, encompassing non-geodesic and non-proper length spaces, and yields a boundary theory extending both the ideal boundary of CAT(0) spaces and the Gromov boundary of hyperbolic spaces, thus providing a unified approach to the bordification of spaces with negative or nonpositive curvature characteristics (Buckley et al., 2012).

1. Definition and Formulation

Let $(X, d)$ be a length space and fix $C \geq 0$. For $h \geq 0$, an $h$-short segment $[x, y]_h$ is a path joining $x$ to $y$ of length $\leq d(x, y) + h$. An $h$-short triangle $T^h(x, y, z)$ consists of three $h$-short sides $[x, y]_h$, $[y, z]_h$, and $[z, x]_h$. Each triangle is compared with a unique Euclidean triangle $\bar T(\bar x, \bar y, \bar z)$, where $|\bar x - \bar y| = d(x, y)$, etc.

Define

$$H(x, y, z) = \frac{1}{1 \vee d(x, y) \vee d(y, z) \vee d(z, x)}.$$

$(X, d)$ is said to be $C$-rough CAT(0) ($C$-rCAT(0)) if, for every triple $x, y, z$ and every $h$-short triangle $T^h(x, y, z)$ with $0 \leq h \leq H(x, y, z)$, the rough CAT(0) inequality holds: for any $u \in [x, y]_h$, $v \in [x, z]_h$ and corresponding $h$-comparison points $\bar u, \bar v$ in the model triangle,

$d(u, v) \leq |\bar u - \bar v| + C.$

A space $X$ is rough CAT(0) (rCAT(0)) if it is $C$-rCAT(0) for some $C \geq 0$.

2. Equivalent Characterizations and Associated Inequalities

2.1 Weak and Metric Characterizations

A weak but equivalent condition (up to normalization of constants) for rCAT(0) asserts that for every $h$-short triangle $T^h(x, y, z)$, if $u$ subdivides $[y, z]_h$ in the ratio $t : (1-t)$,

$$\bigl(d(x, u) - C\bigr)^2 \leq (1-t)d(x, y)^2 + t d(x, z)^2 - t(1-t)d(y, z)^2.$$

This coincides with the classical CAT(0) inequality when $C=0$ but allows for controlled additive error.

Buckley–Hanson showed that for $n \geq 5$, one can replace the triangle comparison with a purely metric "rough $n$-point condition." This generalizes the 4-point inequality used in CAT(0) geometry, now at the cost of increasing the number of points and accommodating additive errors.

2.2 Relations to Other Notions

A very weak form of rCAT(0), considering only comparisons between a point on one side and a vertex, is quantitatively equivalent to the "bolicity" condition introduced by Kasparov–Skandalis. This aligns rCAT(0) with classes of Boltzmann ("bolic") spaces.

A $\delta$-hyperbolic length space is $C$-rCAT(0) with $C = 2 + 4\delta$, so Gromov hyperbolic and CAT(0) spaces both lie within the rCAT(0) class.

3. Fundamental Examples

3.1 Canonical Instances

• Every complete CAT(0) space is $C$-rCAT(0) for any $C > 0$ (arbitrarily small $C$ if the short-segment bound is weakened). Euclidean space and symmetric spaces of nonpositive curvature are included.
• Every Gromov $\delta$-hyperbolic length space is $C$-rCAT(0) with $C = 2+4\delta$.
• Non-geodesic proper length spaces that arise as limits or gluings of CAT(0) pieces, even when not geodesic, remain rCAT(0).
• Certain subspaces of the plane can be CAT(0) with no geodesic rays (empty ideal boundary) but are rCAT(0) with a nontrivial bouquet boundary.

3.2 Table of Example Classes

Space Class	$C$-rCAT(0) Constant	Notable Feature
Complete CAT(0) space	any $C>0$	Admits usual CAT(0) boundary
Gromov $\delta$-hyperbolic length space	$C=2+4\delta$	Admits Gromov boundary
Gluings/limits of CAT(0) spaces	some $C>0$	Can lack geodesicity, yet rCAT(0)
Non-geodesic CAT(0) subspace (bouquet case)	some $C>0$	Empty ideal boundary, rCAT(0) holds

4. Structural Theorems and Fundamental Properties

• Proposition 3.3 (Buckley–Falk): If $X$ is CAT(0), then $X$ is $C$-rCAT(0) with $C=15$; specifically, $X$ is $C$-rCAT(0;*) for all $C > 0$.
• Proposition 3.4: If $X$ is $\delta$-hyperbolic, then $X$ is $(2+4\delta)$-rCAT(0).
• Rough convexity (Lemma 3.6): In a $C$-rCAT(0) space, any pair of $h$-short paths $\alpha,\beta$ from $a_i$ to $b_i$ satisfy

$$d(\alpha(t), \beta(t)) \leq (1-t)d(a_1,a_2) + t d(b_1,b_2) + 2C$$

for all $t\in[0,1]$. This extends the classical convexity property in CAT(0) geometry with an additive error.

A $\delta$-hyperbolic space can be seen as rCAT(0) by employing the 4-point hyperbolicity condition and the tripod lemma, establishing that rCAT(0) subsumes both CAT(0) and hyperbolic spaces.

5. Boundary Theory: Bouquets and Bordification

For classical CAT(0) spaces, the ideal boundary $\partial_I X$ consists of equivalence classes of geodesic rays. rCAT(0) spaces may lack geodesic rays or well-behaved ray structure; thus, Buckley–Falk introduced the theory of bouquets of short paths.

A standard bouquet based at $o \in X$ is a sequence of unit-speed $D$-short paths $B_n : [0, \ell_n] \to X$ ($D(t) = 1/(1 \vee 2t)$, $\ell_n \to \infty$) such that, for $m \leq n$ and $t \leq \ell_m$,

$d(B_m(t), B_n(t)) \leq 2C+2.$

Bouquets $B, B'$ are equivalent if they remain at bounded distance at each $t$. The bouquet boundary $\partial_B X$ is the set of equivalence classes of bouquets.

The boundary theory satisfies:

• For complete CAT(0) spaces, $\partial_B X$ coincides (homeomorphically, in the cone topology) with the usual ideal boundary $\partial_I X$.
• For $\delta$-hyperbolic spaces, $\partial_B X$ coincides with the Gromov boundary $\partial_G X$.
• In any rCAT(0) space $X$, the bordification $X \cup \partial_B X$ can be topologized so that it is Hausdorff, first countable, and $X$ is dense. If $X$ is proper, the bordification is compact.

6. Sketches of Key Proof Strategies

The validation that rCAT(0) generalizes both CAT(0) and Gromov hyperbolic spaces invokes:

• Hyperbolic $\Rightarrow$ rCAT(0): In $\delta$-hyperbolic spaces, any two short segments between the same endpoints stay within the required additive error compared to the Euclidean model, by applying the 4-point hyperbolicity condition and the tripod lemma.
• CAT(0) $\Rightarrow$ weak rCAT(0): Geodesic segments are approximated by $h$-short paths; applying the exact CAT(0) comparison with bounded accumulated error yields the rCAT(0) inequality.
• Rough convexity: The $C$-rough CAT(0) inequality implies that, at proportional times, subpaths on different short sides of a triangle remain within a controlled additive distance, paralleling the usual convexity in CAT(0) theory, adjusted for coarse geometry.
• Boundary convergence: The bouquet boundary is stable under pruning bouquets and changing basepoints, as ensured by the rough CAT(0) and rough convexity properties, making boundary constructions robust within the coarse geometric framework.

7. Context and Significance

The coarse CAT(0) property delineates the minimal natural coarse-geometric class encompassing both CAT(0) and Gromov hyperbolic spaces, demonstrating stability under limiting operations. The bouquet boundary provides a unified bordification at infinity, coinciding with classical boundaries in CAT(0) and hyperbolic extremes, and remaining meaningful even where ideal/geodesic ray boundaries are vacuous. This construction facilitates deeper understanding of boundary phenomena and coarse geometry for spaces with nonpositive or negative curvature traits (Buckley et al., 2012).