Quasi-isometry

A map $f:X\to Y$ is a $(\lambda ,\epsilon )$-quasi-isometric embedding whenIt is a quasi-isometry when every point of $Y$ lies a uniformly bounded distance from its image.

A quasi-isometric embedding satisfies the two-sided coarse distance inequality in the definition of quasi-isometry, without requiring its image to be coarsely dense.

A finitely generated subgroup $H\le G$ is quasi-isometrically embedded when its inclusion, equipped with word metrics from finite generating sets, is a quasi-isometric embedding. This property is independent of those generating sets.

A bilipschitz equivalence distorts all distances by multiplicative constants bounded away from zero and infinity. Every bilipschitz equivalence is a quasi-isometry.

A quasigeodesic is a quasi-isometric embedding of an interval into a metric space.

In a Gromov-hyperbolic geodesic metric space, every quasigeodesic segment stays within a uniformly bounded Hausdorff distance of a geodesic segment with the same endpoints. The bound depends only on the hyperbolicity and quasigeodesic constants.

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

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

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.