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.

For a quasi-isometry $f:X\to Y$ with multiplicative constant $\lambda$ and additive/coarse-surjectivity constant $ϵ$, choose $f^{\prime }(y)$ with $d_Y(f(f^{\prime }(y)),y)\le ϵ$. The triangle inequality and the two distortion inequalities giveAlso $d_X(f^{\prime }(f(x)),x)\le 2\lambda ϵ$. Hence $f^{\prime }$ is a quasi-isometry and both composites stay uniformly close to their respective identities. No continuity is required.

If $H$ is a finite-index subgroup of a finitely generated group $G$, then $H$ is finitely generated and its inclusion into $G$ is a quasi-isometry. Finite generation follows from Schreier's lemma, while finitely many coset representatives give coarse surjectivity and uniformly bounded distortion.

If a group $G$ is generated by $S$, a subgroup $H$ has a set $T$ of right-coset representatives, and $\overline{ts}$ denotes the representative of $Hts$, then $H$ is generated by the Schreier generatorsIn particular, every finite-index subgroup of a finitely generated group is finitely generated.

Given a presentation of a group and a Schreier transversal for a subgroup, the Reidemeister–Schreier theorem gives a presentation of the subgroup. Its generators are the Schreier generators $ts{\overline{ts}}^{-1}$, and its relators are rewritten conjugates of the original relators.

For a subgroup $H$ of a group generated by a set $S$, choose representatives $t$ of right cosets $Ht$. If $\overline{tg}$ represents $Htg$, then $tg{\overline{tg}}^{-1}$ lies in $H$. These Schreier generators generate $H$: inserting successive coset representatives along a word telescopes the word into such factors. Lifted relators can be rewritten in the same factors.

If a finitely generated group $G$ maps surjectively to $H$ with finite kernel, the quotient map is a quasi-isometry. After lifting a finite generating set of $H$, the only possible extra length lies in the finite kernel and is therefore uniformly bounded.

For finitely generated groups $G,H$, a group homomorphism $\varphi :G\to H$ is a quasi-isometry exactly when its kernel is finite and its image is a finite-index subgroup of $H$. The finite kernel controls collapse of distances, and finite index is exactly the coarse-surjectivity condition.

A group is virtually free when it has a free group as a finite-index subgroup. Every finitely generated virtually free group is quasi-isometric to a tree.

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 subgroup $H\le G$ is a retract when there is a group homomorphism $r:G\to H$ whose restriction to $H$ is the identity. If both groups are finitely generated, the inclusion of $H$ is a quasi-isometric embedding: the inclusion and retraction are both Lipschitz in their word metrics.

For a finitely generated subgroup $H\le G$, subgroup distortion compares the intrinsic word length $|h{|}_H$ with the ambient word length $|h{|}_G$. The subgroup is undistorted exactly when its inclusion is a quasi-isometric embedding.

This is the subgroup word-metric instance of distortion.

The central infinite cyclic subgroup of the Integer Heisenberg group has quadratic subgroup distortion. The identity $c^{k^2}=[a^k,b^k]$ gives ambient length at most $4k$ against intrinsic length $k^2$. For general $m\ge 0$, take $k=\lfloor \sqrt{m}\rfloor$ and write $m=k^2+r$, $0\le r\le 2k$, to obtain length at most $4k+r\le 6\sqrt{m}$. Conversely, a word of length $n$ in $a,b,c$ changes the first coordinate by at most $n$ and its central coordinate by at most $n^2+n$, so $|c^m|$ is bounded below by a positive multiple of $\sqrt{|m|}$ for $m\ne 0$. Negative exponents follow by inverses. Thus the ambient and intrinsic metrics have quadratic relative growth.

A subgroup is exponentially distorted when elements with intrinsic length exponential in $n$ can have ambient length linear in $n$. In $\mathrm{BS}(1,2)$, the identity $b^{n}a{b}^{-n}=a^{2^n}$ makes the cyclic subgroup generated by $a$ exponentially distorted.

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.

The bounded-distance conclusion is a property of a Gromov-hyperbolic metric space, under the geodesic and quasigeodesic hypotheses stated here.

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.