Hyperbolic metric space ($\delta$)

For $\delta \ge 0$, a geodesic metric space is $\delta$-hyperbolic when each side of every metric geodesic triangle lies in the closed $\delta$-neighbourhood of the other two sides.

For geodesic metric spaces, a quasi-isometry preserves the property of being a Gromov-hyperbolic metric space. Images of metric geodesic sides are uniform quasigeodesics. The Morse lemma for quasi-geodesics keeps them uniformly close to metric geodesic sides in the hyperbolic target. Thinness there and the coarse lower distance bound pull a uniform triangle-thinness constant back to the source. Coarse surjectivity supplies a quasi-inverse for the reverse implication. Consequently a metric geodesic space quasi-isometric to a tree is hyperbolic, with no local finiteness assumption on the tree.

A geodesic triangle is $\delta$-thin when every point on each side lies within distance $\delta$ of one of the other two sides.

Every side of a geodesic quadrilateral in a $\delta$-hyperbolic metric space lies in the closed $2\delta$-neighbourhood of the other three sides. Indeed, a diagonal divides the quadrilateral into two $\delta$-thin triangles.

A subset $Y$ of a geodesic metric space is convex when every metric geodesic whose endpoints lie in $Y$ is contained in $Y$.

If $Y$ is convex in a $\delta$-hyperbolic metric space and two points of $Y$ both minimize distance from $x$, then their distance is at most $4\delta$. Thus a closest-point projection to $Y$, when it exists, is unique up to uniformly bounded error.

If $\gamma :\mathbb{R}\to X$ is an isometrically embedded geodesic line, then every $x\in X$ has a closest point on $\gamma$. The function $t\mapsto d(x,\gamma (t))$ is continuous and tends to infinity with $|t|$. Any two minimizing parameters differ by at most $4\delta$, and therefore by at most $6\delta$, in a $\delta$-hyperbolic metric space.

For a geodesic triangle with side lengths $a,b,c$, its tripod points divide the three sides into lengthsIn a $\delta$-hyperbolic metric space the three tripod points have pairwise distance at most $4\delta$.

A finitely generated group is hyperbolic when one, equivalently every, Cayley graph for a finite generating set is a Gromov-hyperbolic metric space.

If $g$ has infinite order in a hyperbolic group, the map $n\mapsto g^n$ is a quasi-isometric embedding of the integers. Equivalently $|g^n|\ge c|n|-C$ for some $c>0$. This is the homogeneous quasigeodesic theorem: local-to-global control of metric geodesics in a hyperbolic Cayley graph gives a quasigeodesic orbit for a suitable conjugate of a positive power; undoing conjugation and passing between consecutive powers changes only uniform constants. It implies $\tau (g)>0$. The theorem and its hypotheses are given in Löh, Theorem 6.5.9. It is not true for arbitrary finitely generated groups, as the Baumslag-Solitar group relation $ab{a}^{-1}=b^3$ shows.

A quasi-tree is a geodesic metric space quasi-isometric to a tree.

A geodesic metric space has the bottleneck property if every path joining the endpoints of a geodesic passes within one uniform distance of that geodesic's midpoint. A geodesic metric space is a quasi-tree exactly when it has this property.