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A cobounded group action by isometries on a nonempty geodesic metric space admits an orbit map that is a quasi-isometry for a possibly infinite generating set of a group. If $Gx$ is $R$-dense, take $S=\{g\ne 1:d(x,gx)\le 2R+1\}$. Subdivide a metric geodesic into segments of length at most one and approximate subdivision vertices by orbit points, keeping the endpoints exact. The resulting group increments generate and giveThe first estimate includes zero displacement: a nontrivial stabilizer element has length one. No finiteness assertion about $S$ survives without properness.

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

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.

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.

A metric geodesic triangle consists of three points of a geodesic metric space and a chosen metric geodesic segment between each pair. Multiple choices are allowed, and coincident vertices give degenerate triangles. A Gromov-hyperbolic metric space has a uniform bound on the distance from each side to the other two sides for every such choice. This definition applies to Cayley graphs and trees without tangent vectors or curvature assumptions; it does not invoke the surface angle formula for a geodesic triangle.

If a group has a properly discontinuous group action by isometries that is also a cocompact group action on a nonempty proper metric space that is a geodesic metric space, then the group is finitely generated and every orbit map from the word metric for any finite generating set of a group is a quasi-isometry. Finite stabilizers are permitted.

The Milnor–Švarc lemma applies to an isometric properly discontinuous group action that is also a cocompact group action on a nonempty proper metric space that is a geodesic metric space. It concludes that the group is finitely generated and, for a basepoint $x$ and any finite generating set of a group $S$, the orbit mapHere $d_S$ is the word metric, and a quasi-isometry means a map satisfying two-sided coarse distance bounds with a uniformly coarsely dense image. A free action is not required: finite stabilizers are compatible with proper discontinuity. The properness and cocompactness hypotheses ensure that only finitely many group elements move a chosen basepoint by the bounded amount used to generate the group. Replacing a finite generating set by another does not change the conclusion, by equivalence of finite word metrics.

A subset is a quasiconvex subset with constant $C\ge 0$ if every metric geodesic segment with endpoints in it lies within distance $C$ of it. Explicitly,for every choice of the metric geodesic $[p,q]$. The distance is computed in the ambient geodesic metric space, not in the intrinsic path metric of $N$. One only asks the metric geodesic to stay close to $N$, rather than requiring it to lie in $N$; the latter makes $N$ a convex subset of a geodesic metric space.

The usual convention takes $C\ge 0$. If the printed allowance $C\in \mathbb{R}$ is interpreted literally using the distance inequality, a negative $C$ cannot work for a nonempty $N$, since the constant metric geodesic at a point of $N$ has distance zero. For the empty subset the endpoint condition is vacuous.

A subset $Y$ of a geodesic metric space is $K$-quasiconvex when every metric geodesic between points of $Y$ lies in the closed $K$-neighborhood of $Y$.

A real tree is a metric space $(T,d)$ in which every pair $a,b\in T$ is joined by a unique arc and that arc is isometric to the real interval $[0,d(a,b)]$. Equivalently, it is a geodesic metric space with no nontrivial simple loops.