Equivalent definitions of Gromov hyperbolicity Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible.
I'm happy for the definitions to require some niceness conditions on $X$, such as being geodesic or proper, but please note the conditions needed in your answer.
 A: The super-linear divergence of geodesics condition. Let $X$ be a geodesic metric space. A map $e\colon \mathbb{N} \to \mathbb{R}$ is a divergence function for $X$ if for all $R$, $r$ in $\mathbb{N}$, all $x \in X$ and all geodesics $\gamma\colon [0,a_1]\to X$ and $\gamma'\colon [0,a_2] \to X$ with $\gamma(0) = \gamma'(0) = x$ such that $R + r \le \min\{a_1,a_2\}$ and $d(\gamma(R),\gamma'(R)) > e(0)$, then we have that any path connecting $\gamma(R+r)$ to $\gamma'(R+r)$ outside the ball $B(x,R+r)$ has length at least $e(r)$.
We say $X$ is hyperbolic if $X$ has a divergence function $e$ such that $\lim\inf_{n\to\infty}\frac{e(n)}n = \infty$.
Source: Definition III.H.1.24 and Proposition III.H.1.26 in Metric spaces of non-positive curvature by Bridson and Haefliger.
A: The four-point condition. Let $(X,d)$ be a metric space, with $x$, $y$, $z \in X$. The Gromov product $(x,y)_z$ of $x$ and $y$ with respect to $z$ is defined to be the quantity
$$(x,y)_z = \frac{1}{2}\left(d(z,x) + d(z,y) - d(x,y)\right).$$
Given $\delta \ge 0$, the space $X$ is $\delta$-hyperbolic if for all $x$, $y$, $z$ and $w \in X$, we have
$$(x,z)_w \ge \min\{(x,y)_w,(y,z)_w\} - \delta.$$
One can show that if $X$ satisfies the same condition above except with the point $w$ fixed, then $X$ is $2\delta$-hyperbolic in the sense of the approximate incenter condition. Note that this definition does not require $X$ to be geodesic.
Source: Section 1.1 of Hyperbolic groups by Gromov.
A: The slim triangles condition. Let $X$ be a geodesic metric space and $\delta \ge 0$. Given two points $x$ and $y \in X$, let $[x,y]$ denote a geodesic between them. We say $X$ is $\delta$-hyperbolic if for any three points $x$, $y$ and $z \in X$, we have that any geodesic triangle with vertices $x$, $y$ and $z$ is $\delta$-slim in the sense that
$$N_\delta([x,y]\cup[y,z]) \supset [x,z],$$
where $N_r(S)$ denotes the $r$-neighborhood of the set $S$ in $X$.
Source: See Definition 1.3 of Notes on word hyperbolic groups by Brady et al.  This definition is usually attributed to Eliyahu Rips.
A: The linear isoperimetric inequality condition. Let $X$ be a connected graph.  Suppose that $D$ is a cellulation of the two-disk.  We define $A(D)$ to be the number of two-cells of $D$.  We say that $D$ is $n$-coarse if the length of the boundary of any two-cell of $D$ is at most $n$.  Suppose that $f : D^{(1)} \to X$ is a graph map (sending vertices to vertices and edges to edges).  We say that $(D, f)$ spans the loop $f(\partial D)$ in $X$.  Finally, for a loop $\gamma$ in $X$ we define $L(\gamma)$ to be the number of edges of $\gamma$.
We say that $X$ is hyperbolic if there are positive constants $n$, $a$, and $b$ with the following property.  For any edge loop $\gamma$ in $X$, there is an $n$-coarse disk $(D, f)$ spanning $\gamma$ so that $A(D) \leq a \cdot L(\gamma) + b$.
Source: Proposition 3.1 of Relatively hyperbolic groups by Bowditch.
A: The exponential divergence of geodesics condition.
Let $(X,d)$ be a geodesic metric space, with some basepoint $x_0$. Take $X_r = X\setminus B(r)$ and define the circumferential distance on $X_r$ given by $d_r(x,y) = \inf_{\gamma}\{\text{len } \gamma,\infty\}$ where $\gamma$ ranges over all curves connecting $x$ and $y$, and contained in $X_r$.
(In the case of the hyperbolic plane, and $x$ and $y$ are distance $r$ from $x_0$ then $m_r(x,y)$ is the length of the shorter circular arc centered at $x_0$ which connects them.)
Geodesics from $x_0$ diverge exponentially (K), if the follow condition is satisfied:
Let $\ell_1,\ell_2$ denote geodesic rays centered at $x_0$ and let $r$ and $s$ be any positive numbers satisfying $0<r<r+K\leq s$. If $a_1,b_1\in \ell_1$ and $a_2,b_2\in \ell_2$, with $d(x_0,a_1)=r=d(x_0,a_2)$ and $d(x_0,b_1)=s=d(x_0,b_2)$, then
$$d_s(b_1,b_2)\geq 2d_r(a_1,a_2) - 2K.$$
This condition is a way of saying that for coarsely hyperbolic spaces, the circumference of circles grows exponentially (since adding a fixed constant $K$ to the radius approximately doubles the circumferential distance).
Having exponentially divergent geodesics (K) and $\delta$-thin triangles are (qualitatively) the same, and the source includes some estimates relating them.
Source: The theory of negatively curved spaces and groups by James W. Cannon, in Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces.
A: The short chords condition. Let $X$ be a connected graph.  For any edge-path or edge-loop $\gamma$ in $X$, we use $L(\gamma)$ to denote the number of edges in $\gamma$.
Suppose that $\gamma$ is an embedded loop in $X$.  A path $\alpha$ in $X$ is a chord for $\gamma$ if $\alpha$ starts and ends on distinct vertices of $\gamma$.  A chord $\alpha$ for $\gamma$ is well-separated if each arc of $\gamma - \partial \alpha$ contains at least $L(\gamma)/4$ edges.
We say that $X$ is hyperbolic if there is a positive constant $K$ with the following property:  every edge-loop $\gamma$ in $X$, with $L(\gamma) \geq K$, admits a well-separated chord $\alpha$ where $L(\alpha) \leq L(\gamma)/6 + K$.
Source: Based on Theorem B of The geometry of cycles in the Cayley diagram of a group by Gilman.
A: The thin bigons condition. Let $X$ be a graph, given the graph metric (all edges have length $1$). We say $X$ is hyperbolic if there exists $\epsilon > 0$ such that for every pair of geodesic segments $\gamma$ and $\gamma'$ in $X$ with the same endpoints we have that $N_\epsilon(\gamma)$, the $\epsilon$-neighborhood of $\gamma$, contains $\gamma'$ and similarly $N_\epsilon(\gamma') \supset \gamma$.
Source: Theorem 1.4 of Strongly geodesically automatic groups are hyperbolic by Papasoglu.
A variant: the thin $q$-bigons condition. Let $X$ be a geodesic metric space. Recall that a $(1,q)$-quasigeodesic segment is a function $\gamma\colon [a,b] \to X$ where $[a,b]$ is a closed interval of $\mathbb{R}$ with the property that for all $s$ and $t$ in $[a,b]$, we have
$$|s-t| - q \le d(\gamma(s),\gamma(t)) \le |s-t| + q.$$
A $q$-bigon in $X$ is a pair of $(1,q)$-quasigeodesic segments $\gamma$ and $\gamma'$ with the same endpoints.
We say $X$ is hyperbolic if there exists an $\epsilon > 0$ and a $q > 0$ such that all $q$-bigons $\gamma$ and $\gamma'$ satisfy $N_\epsilon(\gamma) \supset \gamma'$ and $N_\epsilon(\gamma') \supset \gamma$.
Source: Pomroy's Master's thesis; given a new proof as Theorem 22 of A characterization of hyperbolic spaces by Chatterji and Niblo
A: The approximate incenter condition. Let $X$ be a geodesic metric space and $\delta \ge 0$. Given two point $x$ and $y$ in $X$, let $[x,y]$ denote a geodesic between them. Given a geodesic triangle with vertices $x$, $y$ and $z$, a $C$-center of the triangle is a point $p$ such that $d(p,[x,y]) \le C$ and $d(p,[x,z]) \le C$ and $d(p,[y,z]) \le C$. The space $X$ is $\delta$-hyperbolic if every geodesic triangle admits a $\delta$-center.
A: The thin triangles condition. Let $X$ be a geodesic metric space and $\delta \ge 0$. Given two points $x$ and $y \in X$, let $[x,y]$ denote a geodesic between them.  Suppose that $x$, $y$ and $z$ are points of $X$.  Let $\Delta$ be a collection of three geodesics collecting them.  There are points $c_z \in [x, y]$, $c_y \in [z, x]$, and $c_x \in [y, z]$ so that $d(x, c_y) = d(x, c_z)$ and similarly for $y$ and $z$.  Now, there is a unique (up to marked isometry) tripod (metric tree) $\Delta'$ with leaves $x'$, $y'$, and $z'$ and interior vertex $c$ so that $d(x', c) = d(x, c_y)$ and so on.
There is also a unique local isometry $f : \Delta \to \Delta'$. We say that $\Delta$ is $\delta$-thin if the fibers of $f$ have diameter at most $\delta$ in $X$.
We say $X$ is $\delta$-hyperbolic if all geodesic triangles are $\delta$-thin.
Source: Definition 1.5 of Notes on word hyperbolic groups by Brady et al.
A: The strong contraction condition. Let $X$ be a geodesic metric space. Given an interval $I \subset \mathbb R$ and a path $\gamma : I \to X$, and given constants $a,b,c>0$, to say that $\gamma$ has the $a,b,c$-strong contraction property means that there exists a map $\pi : X \to I$ such that:

*

*For any $t \in I$, the diameter of $\gamma[t,\pi(\gamma(t))]$ is at most $c$.

*For any $x,y \in X$, if $d(x,y) \le 1$ then the diameter of $\gamma[\pi(x),\pi(y)]$ is at most $c$.

*For any $x \in X$, if $d\bigl(x,\gamma(\pi(x))\bigr) \ge a$ and if $d(x,y) \le b \cdot d\bigl(x,\gamma(\pi(x))\bigr)$ then the diameter of $\gamma[\pi(x),\pi(y)]$ is at most $c$.

(Interval notation $[s,t]$ is used symmetrically).
Given $d > 0$, a set of paths in $X$ is $d$-coarsely transitive if for all $x,y \in X$ with $d(x,y) \ge d$ there is a path in the family passing through $x$ and $y$.
$X$ is hyperbolic if and only if there exist constants $a,b,c,d>0$ and a $d$-coarsely transitive set of paths each of which satisfies the $a,b,c$-strong contraction property.
Source: Definition 2.2 and Theorem 2.3 of Geometry of the complex of curves I: Hyperbolicity by Masur and Minsky.
A: The line-center conditions. Let $X$ be a connected graph with path metric assigning length $1$ to each edge.
For any vertices $a,b \in X$ we are given a line from $a$ to $b$, which is a subset $\Lambda_{ab} \subset X$ containing $a,b$ equipped with a coarse order $\le_{ab}$ meaning a reflexive, transitive relation satisfying the dichotomy law $x \le_{ab} y$ or $y \le_{ab} x$, for all $x,y \in \Lambda_{ab}$, and such that $a$ is a minimum and $b$ is a maximum in $\Lambda_{ab}$. Given $x \le_{ab} y$ in $\Lambda_{ab}$ the subinterval $\Lambda_{ab}[x,y]$ is the set of all $z \in \Lambda_{ab}$ such that $x \le_{ab} z \le_{ab} y$.
We are also given a function $\phi : X \times X \times X \to X$ that assigns to each $(a,b,c)$ a point $\phi(a,b,c) \in \Lambda_{ab} \cap \Lambda_{bc} \cap \Lambda_{ca}$ that is called the center of $a,b,c$.
We assume that the following conditions hold for some constant $K \ge 0$ and for all vertices $a,b,c,x,y$:

*

*$\phi(a,b,c)=\phi(b,a,c)=\phi(a,c,b)$ and $\phi(a,a,b)=a$.

*The sets $\Lambda_{ab}[a,\phi(a,b,c)]$ and $\Lambda_{ac}[a,\phi(a,b,c)]$ have Hausdorff distance $\le K$.

*If $d(x,y) \le 1$ then the set $\Lambda_{ab}[\phi(a,b,x),\phi(a,b,y)]$ has diameter $\le K$.

*If $c \in \Lambda_{ab}$ then the set $\Lambda_{ab}[c,\phi(a,b,c)]$ has diameter $\le K$.

Under these conditions we say that $X$ is hyperbolic.
Source: Proposition 3.1 in Section 3 of "Intersection numbers and the hyperbolicity of the curve complex", by Bowditch.
Notice that lines are unparameterized, and are not assumed to be geodesics or quasigeodesics of any kind. In fact an additional conclusion of Proposition 3.1 says that each $\Lambda_{ab}$ has uniform Hausdorff distance from each geodesic with endpoints $a,b$.
A: The intersections of balls are close to balls condition.
Let $X$ be a geodesic metric space. We say that $X$ is hyperbolic if the intersection of metric balls is at uniformly bounded Hausdorff distance from a ball.
Source: Theorem 1 of "A characterization of hyperbolic spaces" by Chatterji and Niblo.
A: The uniformly bounded eccentricity of intersections of balls condition. Let $X$ be a geodesic metric space. We say a subset $S$ of $X$ has eccentricity less than $\delta$ for some $\delta \ge 0$ if there exists $R \ge 0$ such that
$$B(c,R) \subset S \subset B(c’,R + \delta)$$
for some points $c$ and $c’$ in $X$. (By convention the empty set has eccentricity $0$.) Here of course $B(c,R)$ denotes the ball of radius $R$ centered at $c$. We say that $X$ is hyperbolic if there exists a uniform $\delta$ such that the intersection of any two metric balls has eccentricity at most $\delta$.
Source: Proposition 14 and Lemma 17 of Chatterji and Niblo, “A characterization of hyperbolic spaces”
A: The space of triples condition. The following is a fabulous condition for a group $\Gamma$ to be hyperbolic. It has no mention of a metric on $\Gamma$ whatsoever. Nevertheless, I think it fits this list.
Suppose that $M$ is a perfect metrisable compactum. Suppose that
a group $\Gamma$ acts by homeomorphism on $M$ such that the induced action on the space of
distinct triples is properly discontinuous and cocompact. Then, $\Gamma$ is hyperbolic. Moreover,
there is a $\Gamma$-equivariant homeomorphism of $M$ onto $\partial\Gamma$.
Source: A topological characterisation of hyperbolic groups by Brian Bowditch. The condition is given as the article's main theorem.
A: The tripod condition A geodesic space is $\delta$-hyperbolic, $\delta \geq 0$, if for
any triangle $xyz \subset X$ the following holds: If $y' \in xy, z' \in xz$ are points with $|xy'| = |xz'| \leq (y|z)_x$, then $|y'z'| \leq \delta$.
Source: Definition 1.1.2. of Buyalo and Schroeder "Elements of Asymptotic Geometry".
A: The slim subgraph condition. Suppose that $X$ is a graph where all edges have length one. Then $X$ is Gromov hyperbolic if and only if there is a constant $M \geq 0$ and there is, for all (unordered) pairs of vertices $x$ and $y$ in $X$, a connected subgraph $g_{x,y} \subset X$ containing $x$ and $y$, with the following properties.

*

*(Local) If $d_X(x, y) \leq 1$, then $g_{x,y}$ has diameter at most $M$.

*(Slim triangles) For all vertices $x$, $y$, and $z$ \in $X$ the subgraph $g_{x,y}$ is contained in an $M$–neighborhood of $g_{y,z} \cup g_{z,x}$.

Source: Theorem 3.15 of Masur and Schleimer "The geometry of the disk complex".
A: The divergent geodesic condition. (This is a consequence of being hyperbolic but I am not sure if it is an equivalent condition). Say that geodesics $\gamma_1$ and $\gamma_2$ are $M$-close if for any $x$ on $\gamma_i$ there exists a $y$ on $\gamma_j$ with $d(x,y)<M$.
If $(X,d)$ has $\delta$-thin triangles then there is some $M_0$ such that no two geodesics are $M_0$-close.
