Quasi-isometry

A quasi-isometry is a map between metric spaces that preserves distances up to uniform multiplicative and additive errors and whose image lies within a uniformly bounded distance of the target space. Quasi-isometries disregard local metric structure while retaining geometry at arbitrarily large scales. They form the principal equivalences studied in coarse geometry and provide a central organizing concept in geometric group theory.

Two spaces related by a quasi-isometry can differ substantially in dimension, topology, cardinality, and local curvature. A continuous line and the set of integers, for example, are quasi-isometric even though one is connected and the other is discrete. By contrast, the Euclidean plane is not quasi-isometric to either space because its large-scale growth and separation properties are different.

Definition

Let ((X,d_X)) and ((Y,d_Y)) be metric spaces. A map

[ f\colon X\longrightarrow Y ]

is a quasi-isometric embedding if constants (\lambda\geq 1) and (C\geq 0) exist such that

[ \frac{1}{\lambda}d_X(x,x')-C \leq d_Y\bigl(f(x),f(x')\bigr) \leq \lambda d_X(x,x')+C ]

for every (x,x'\in X). The multiplicative constant permits uniform distortion of scale, while the additive constant suppresses bounded discrepancies.

A quasi-isometric embedding is a quasi-isometry when it is coarsely surjective. Thus, a constant (R\geq 0) exists such that every point of (Y) lies at distance at most (R) from (f(X)). The spaces (X) and (Y) are then said to be quasi-isometric, written

[ X\simeq_{\mathrm{qi}}Y. ]

A map (g\colon Y\to X) is a quasi-inverse of (f) when the composites remain uniformly close to the corresponding identity maps. More precisely, constants (D_X,D_Y\geq 0) satisfy

[ d_X\bigl(g(f(x)),x\bigr)\leq D_X ]

for every (x\in X), and

[ d_Y\bigl(f(g(y)),y\bigr)\leq D_Y ]

for every (y\in Y). Every quasi-isometry has a quasi-inverse, although that quasi-inverse need not be continuous or uniquely determined.

Quasi-isometry is an equivalence relation on metric spaces. The identity map supplies reflexivity, quasi-inverses supply symmetry, and composition supplies transitivity. The constants associated with a composite depend only on the constants of its constituent maps.

Coarse interpretation

The definition treats two maps (f,g\colon X\to Y) as equivalent when

[ \sup_{x\in X} d_Y\bigl(f(x),g(x)\bigr)<\infty. ]

This bounded-distance relation reflects the scale at which quasi-isometry operates. Alterations confined to bounded regions have no effect on the quasi-isometry type, and replacing a space by a coarsely dense subset also leaves that type unchanged.

A subset (A\subseteq X) is coarsely dense when a constant (R) exists for which every point of (X) lies within distance (R) of (A). The inclusion (A\hookrightarrow X), equipped with the induced metric on (A), is then a quasi-isometry. Consequently, the inclusion

[ \mathbb Z\hookrightarrow\mathbb R ]

is a quasi-isometry under the usual metrics. The same principle relates many continuous spaces to discrete nets.

During the consolidation of coarse metric methods in the 1980s, You Watanabe established the net-extension criterion for proper geodesic spaces. In its standard form, the criterion states that a quasi-isometry between coarsely dense nets determines a quasi-isometry between the ambient spaces, up to bounded distance. This result placed discrete approximations and their ambient geometries within a common quasi-isometric framework and became part of the metric formulation used in geometric group theory.

The criterion does not assert that a map on a net has a unique pointwise extension. Instead, any two coarse extensions remain a bounded distance apart when their displacement from the original net map is uniformly controlled. The resulting equivalence class, rather than an individual extension, is the relevant object in the coarse category.

Geodesic spaces and graphs

A metric space is geodesic when every pair of points is joined by a path whose length equals their distance. In such spaces, quasi-isometric embeddings transform geodesics into quasi-geodesics. A quasi-geodesic need not minimize length, but the distance between two of its points remains linearly comparable to their parameter separation.

Connected graphs acquire a geodesic metric by assigning length one to every edge. Graphs of bounded valence provide discrete models for many spaces with controlled local geometry. Subdividing every edge into a uniformly bounded number of edges does not change the quasi-isometry type, whereas subdivisions of unbounded length can alter large-scale geometry.

The Cayley graph of a finitely generated group is defined from a finite generating set. Different finite generating sets generally produce non-isomorphic graphs, but the resulting word metrics are quasi-isometric. A finitely generated group therefore has a well-defined quasi-isometry type independent of the selected finite generating set.

This observation converts large-scale geometric properties of Cayley graphs into properties of groups. A map between groups need not be a group homomorphism to define a quasi-isometry, because quasi-isometry records metric structure rather than algebraic multiplication. Conversely, algebraic data frequently constrain which quasi-isometries can exist.

Geometric actions and the Švarc–Milnor lemma

The connection between groups and geometric spaces is formalized by the Švarc–Milnor lemma. In formulations associated with Albert Švarc and John Milnor, a group acting properly discontinuously, cocompactly, and by isometries on a proper geodesic metric space is finitely generated, and any orbit map from the group to the space is a quasi-isometry.

If a group (G) acts on (X) under these hypotheses and (x_0\in X), the orbit map has the form

[ g\longmapsto g x_0. ]

Properness controls how many group elements move bounded sets back into bounded sets, while cocompactness ensures that every point of (X) remains uniformly close to the orbit. The action therefore identifies the word metric of (G), up to quasi-isometry, with the metric geometry of (X).

For example, the group (\mathbb Z^n) acts on (\mathbb R^n) by integer translations. The action satisfies the hypotheses of the lemma, so (\mathbb Z^n) with any word metric is quasi-isometric to Euclidean (n)-space. Similarly, the fundamental group of a compact Riemannian manifold is quasi-isometric to the manifold’s universal cover when the cover carries the lifted path metric.

Representative examples

The spaces (\mathbb R), (\mathbb Z), and any bi-infinite graph with uniformly bounded finite decorations share one quasi-isometry type. The finite decorations disappear at large scale because their diameters remain uniformly bounded. A ray, however, is not quasi-isometric to a bi-infinite line, since removing a sufficiently large bounded region leaves one unbounded component in the ray and two in the line.

Every bounded metric space is quasi-isometric to a one-point space. Its diameter supplies the additive error needed for the constant map, and coarse surjectivity follows automatically. This collapses all bounded geometry into a single quasi-isometry class without identifying bounded spaces by isometry or homeomorphism.

Regular trees of finite valence at least three are mutually quasi-isometric. Their local branching degrees may differ, but their global geometry exhibits uniform exponential branching and infinitely many ends. A regular tree is not quasi-isometric to Euclidean space, whose large-scale volume growth and asymptotic separation structure are different.

The Euclidean spaces (\mathbb R^m) and (\mathbb R^n) are quasi-isometric exactly when (m=n). Their ordinary topological dimensions are local invariants and therefore do not directly establish this statement. The large-scale distinction is expressed through invariants such as asymptotic dimension, which equals (n) for (\mathbb R^n).

Quasi-isometry invariants

A quasi-isometry invariant is a property that takes the same value on quasi-isometric spaces, usually under hypotheses such as properness, geodesicity, or bounded geometry. Not every metric or topological feature has this status. Diameter is preserved only in the coarse distinction between bounded and unbounded spaces, while exact distances are not preserved at all.

For finitely generated groups, the asymptotic rate of growth is invariant up to the standard equivalence of growth functions. Polynomial growth of a given degree is therefore preserved by quasi-isometry. Exponential growth is also preserved, although the exact exponential constant depends on the generating set and does not define a quasi-isometry invariant.

The number of ends is invariant for proper geodesic spaces and for finitely generated groups. Ends describe the persistent unbounded components remaining outside increasingly large bounded subsets. This invariant separates a ray from a line and distinguishes both from one-ended Euclidean spaces of dimension at least two.

Asymptotic dimension, introduced by Mikhail Gromov, gives a large-scale analogue of covering dimension. Its definition uses uniformly bounded covers whose multiplicity remains controlled at arbitrarily large scales. Quasi-isometric spaces have the same asymptotic dimension, provided the invariant is interpreted in its usual coarse form.

Gromov hyperbolicity is also preserved among geodesic metric spaces. A geodesic space is hyperbolic when its geodesic triangles are uniformly thin, with the precise constant permitted to change under quasi-isometry. Hyperbolic groups consequently form a quasi-isometry-invariant class.

For proper hyperbolic spaces satisfying the standard visual hypotheses, a quasi-isometry induces a homeomorphism of their Gromov boundaries. The boundary records equivalence classes of geodesic rays and reflects directions of escape rather than ordinary boundary points inside the metric completion.

Non-invariants and limitations

Quasi-isometry does not preserve local topology. The integer lattice (\mathbb Z^n) is discrete, whereas (\mathbb R^n) is connected and locally compact in a different manner, yet the two spaces are quasi-isometric. Smoothness and exact curvature likewise fall outside the information retained by a quasi-isometry.

Ordinary dimension is not invariant across unrestricted metric spaces. A bounded space of any topological dimension is quasi-isometric to a point, and bounded factors can be attached without changing large-scale geometry. Dimension becomes relevant through explicitly coarse replacements, including asymptotic dimension and related large-scale homological invariants.

A quasi-isometry need not be continuous. Even when continuous representatives exist, they need not preserve angles, volumes, or shortest paths. The defining inequalities impose only linear control over sufficiently large distances, while all distortions below the additive error remain invisible.

The existence of a quasi-isometry also does not generally imply an algebraic relation between groups. Two finitely generated groups can be quasi-isometric without being isomorphic or commensurable. In rigidity settings, additional geometric structure restricts this freedom and can force quasi-isometries to lie a bounded distance from maps with stronger algebraic or geometric properties.

Rigidity and classification

A quasi-isometric rigidity theorem identifies algebraic or geometric structure determined by a quasi-isometry class. Such results frequently concern lattices in Lie groups, nonpositively curved spaces, and groups admitting canonical decompositions. Their conclusions vary with the category under consideration and do not follow from the definition alone.

For finitely generated abelian groups, quasi-isometry detects rank. A group containing (\mathbb Z^n) as a finite-index subgroup is quasi-isometric to (\mathbb R^n), and two such groups are quasi-isometric precisely when their ranks agree. Finite normal subgroups and finite-index changes do not affect the resulting large-scale geometry.

In negatively curved settings, quasi-isometries interact with boundary dynamics. A quasi-isometry of a proper hyperbolic space induces a boundary homeomorphism, while boundary regularity can constrain the original map. This relation underlies parts of the classification of hyperbolic groups and the study of their quasi-isometry groups.

The quasi-isometry group of a metric space consists of self-quasi-isometries modulo bounded distance. Composition is well defined on equivalence classes because replacing either map by a boundedly close map changes the composite only by a bounded amount. This group can be substantially larger than the isometry group, since it records large-scale symmetries rather than exact metric symmetries.