Quasi-Isometric Invariants

• Quasi-isometric invariants are defined properties or structures of metric spaces that remain unchanged under mappings that distort distances by uniform multiplicative and additive constants.
• They enable large-scale classification by quantifying features like the number of ends, growth rates, Poincaré constants, and asymptotic dimensions across diverse mathematical settings.
• Analytic, combinatorial, and topological methods—including JSJ decompositions and coset intersection complexes—provide practical tools for understanding rigidity and classification in geometric group theory.

A quasi-isometric invariant is a property or structure of a metric space (group, complex, graph, manifold, etc.) that is preserved under quasi-isometry: a map distorting distances only up to uniform multiplicative and additive constants. These invariants provide the main tools for the large-scale classification of spaces and groups up to coarse geometry, and underlie the theory of quasi-isometric rigidity. Their nature and computation vary across geometric group theory, low-dimensional topology, and geometric analysis.

1. Rigorous Definition and Basic Examples

Two metric spaces $(X, d_X)$, $(Y, d_Y)$ are quasi-isometric if there exists $f: X \to Y$ and constants $L \geq 1$, $C \geq 0$ such that

$$\frac{1}{L} d_X(x,x') - C \leq d_Y(f(x), f(x')) \leq L d_X(x,x') + C$$

for all $x, x' \in X$, and such that $Y$ is contained in the $C$-neighborhood of $f(X)$. A quasi-isometric invariant is any property $(Y, d_Y)$0 such that $(Y, d_Y)$1 has $(Y, d_Y)$2 if and only if $(Y, d_Y)$3 has $(Y, d_Y)$4 whenever $(Y, d_Y)$5 and $(Y, d_Y)$6 are quasi-isometric.

Fundamental examples include:

• Number of ends of a finitely generated group or space.
• Asymptotic dimension and Assouad–Nagata dimension.
• Growth rate of balls.
• Divergence function, Dehn function (for groups), and various notions of accessibility.
• Existence (and data) of splittings: e.g., the graph-of-groups decomposition or JSJ decomposition for 3-manifold groups (Frigerio, 2016, Abbott et al., 2024).

2. Invariants in Group-Theoretic and Geometric Topology Contexts

Group Pairs and Peripheral Structures

For pairs $(Y, d_Y)$7 where $(Y, d_Y)$8 is a finitely generated group and $(Y, d_Y)$9 is a finite collection of subgroups (typically reflecting a peripheral structure or decomposition), the quasi-isometry of pairs is defined using the induced Hausdorff geometry on coset spaces. In this framework, the concept of a qi-characteristic collection is central: $f: X \to Y$0 is qi-characteristic if for every quasi-isometry of $f: X \to Y$1, cosets of $f: X \to Y$2 are permuted up to bounded Hausdorff distance (Martínez-Pedroza et al., 2020, Hughes et al., 2021). Invariants in this setting include:

• The quasi-isometry types of the peripheral subgroups.
• The filtered ends $f: X \to Y$3 for $f: X \to Y$4, encoding the large-scale connectedness of $f: X \to Y$5 relative to $f: X \to Y$6.
• The relative Dehn function $f: X \to Y$7, whose growth type is a quasi-isometry invariant (Hughes et al., 2021).
• The relative hyperbolicity of $f: X \to Y$8 with respect to $f: X \to Y$9 (Hughes et al., 2021, Martínez-Pedroza et al., 2020).

JSJ Decompositions and Piecewise Geometric Manifolds

In the context of 3-manifolds and other nonpositively curved spaces, the JSJ decomposition (splitting the group or space along certain subgroups or flats) yields invariants:

• The coarse structure of the Bass-Serre tree underlying the splitting (Frigerio, 2016).
• The quasi-isometry types of vertex (piece) groups.
• Patterns of peripheral subgroups and how quasi-isometries act on them.
• Asymptotic cone structure (tree-graded spaces, presence of flat subspaces) (Frigerio, 2016).

These are encoded combinatorially for group pairs $L \geq 1$0 or via trees of cylinders and related constructions (Cashen, 2014, Abbott et al., 2024).

3. Combinatorial and Homotopical Encodings: Intersection Complexes and Coset Intersection Complexes

A powerful unifying method is to use simplicial complexes or complexes of groups encoding intersection patterns among distinguished subgroups or subspaces. Two key constructions:

• Intersection complex $L \geq 1$1: For the universal cover of a weakly special square complex $L \geq 1$2, the intersection complex $L \geq 1$3 records maximal standard product subcomplexes and their intersections. Its semi-isomorphism class is a quasi-isometry invariant, sufficient to classify 2-dimensional RAAGs in several cases (Oh, 2020).
• Coset intersection complex $L \geq 1$4: For group pairs, $L \geq 1$5 is the flag simplicial complex with vertices the left cosets $L \geq 1$6 and simplices corresponding to cosets with infinite mutual intersection. Properties of $L \geq 1$7 (height, width, almost malnormality, networks) correspond to coarse geometric or topological properties of $L \geq 1$8. Both the metric type and homotopy type of $L \geq 1$9 are quasi-isometry invariants of the pair (Abbott et al., 2024).

Such complexes provide a dictionary translating algebraic or geometric features to purely combinatorial invariants, all preserved under quasi-isometry by construction.

4. Quantitative and Analytic Invariants

Several invariants are defined analytically or via metric measure theory:

• Poincaré Constants and LP Cohomology: $C \geq 0$0 Poincaré inequalities (and associated best constants $C \geq 0$1) are preserved under quasi-isometry up to uniform multiplicative change (Shchur, 2014). The critical LP exponent for first cohomology, $C \geq 0$2, is a quasi-isometry invariant in Gromov-hyperbolic settings.
• Volume Growth and Distortion-Growth Functions: Ball-volume growth type and quantitative lower bounds for distortion under quasi-isometric embeddings are preserved; sharp lower bounds are derived from differences in LP-exponents or exponential versus polynomial growth (Shchur, 2014).
• Homotopy Distortion: Lower and upper bounds for quasi-isometric distortion growth (measured by $C \geq 0$3) are sharp and depend linearly on the difference of certain LP-exponents in classes of twisted product spaces. Sublinear growth can occur in certain Gromov-hyperbolic spaces for suitable boundary maps (Shchur, 2014).

5. Classification of Specific Classes and Algebraic Characterizations

Right-angled Artin Groups and Graph Products

• For RAAGs, the extension graph $C \geq 0$4, and its induced-subgraph embedding relations, serve as the primary quasi-isometry invariant in several classes (trees, atomic graphs, etc.). For atomic RAAGs, graph isomorphism and the co-Hopfian property of the $C \geq 0$5–completion are both quasi-isometry invariants (Casals-Ruiz, 2018).
• For 2-dimensional RAAGs, intersection complexes derived from the universal cover of the Salvetti complex, and their combinatorial type, encode the quasi-isometry classification in the tree and finite outer automorphism cases (Oh, 2020).

Solvable and Nilpotent Lie Groups (Heintze Groups)

• In purely real Heintze groups $C \geq 0$6, the characteristic polynomial (spectrum) of $C \geq 0$7 (up to positive scaling) and, in the Heisenberg case, the full Jordan form (up to scaling), are quasi-isometry invariants (Piaggio et al., 2016).
• Further invariants are described by the associated Carnot-graded Lie algebra (Pansu's asymptotic cone), $C \geq 0$8-cohomology, reachability sets $C \geq 0$9 defined via Hausdorff dimension of curves, and the chain of normalizers derived from these sets. The topological dimension of the asymptotic cone and the isomorphism class of the real-shadow are also quasi-isometry invariants (in low dimensions the real shadow coincides with the QI class) (Kivioja et al., 2021).

6. Comprehensive Table of Quasi-Isometric Invariants by Context

Context	Invariant Example	Source arXiv id(s)
All groups/spaces	Number of ends, growth, asymptotic dim	(Frigerio, 2016, Davies, 22 Oct 2025)
Group pairs $$\frac{1}{L} d_X(x,x') - C \leq d_Y(f(x), f(x')) \leq L d_X(x,x') + C$$0	Filtered ends, peripheral QI types, Dehn fn.	(Martínez-Pedroza et al., 2020, Hughes et al., 2021)
3-manifold groups	JSJ/Bass–Serre tree, piece QI types	(Frigerio, 2016)
RAAGs	Extension graph embeddings, intersection cx.	(Casals-Ruiz, 2018, Oh, 2020)
CAT(0) cube complexes/square complexes	Intersection complex $$\frac{1}{L} d_X(x,x') - C \leq d_Y(f(x), f(x')) \leq L d_X(x,x') + C$$1	(Oh, 2020)
Coset-peripheral pairs	Coset intersection complex $$\frac{1}{L} d_X(x,x') - C \leq d_Y(f(x), f(x')) \leq L d_X(x,x') + C$$2	(Abbott et al., 2024)
Heintze/Solv. Lie groups	Char. polynomial/Jordan, Carnot cone	(Piaggio et al., 2016, Kivioja et al., 2021)
Gromov–hyperbolic spaces	LP-critical exponent $$\frac{1}{L} d_X(x,x') - C \leq d_Y(f(x), f(x')) \leq L d_X(x,x') + C$$3	(Shchur, 2014)
General graphs	Accessibility, planarity, group type	(Davies, 22 Oct 2025)

7. Open Problems and Future Directions

Central open conjectures include:

• Whether the quasi-isometry classification of purely real Heintze groups reduces to isomorphism ($$\frac{1}{L} d_X(x,x') - C \leq d_Y(f(x), f(x')) \leq L d_X(x,x') + C$$4 up to scaling and conjugacy), i.e., whether spectrum, Carnot-graded algebra, and reachability sets suffice in all dimensions (Kivioja et al., 2021, Piaggio et al., 2016).
• Characterization of qi-characteristic collections beyond current classes; unified coarse-invariants for Artin and Coxeter pairs (Hughes et al., 2021).
• Coarse classification of general solvable and nilpotent groups up to quasi-isometry, conjecturally via Carnot-type structure (Kivioja et al., 2021).
• Precise relationships between filtered ends, Dehn functions, and other homological or cohomological invariants for higher complexity pairs (Hughes et al., 2021).
• In extended setups (e.g., general metric complexes or groupoids), the search for new combinatorial or analytic invariants that detect quasi-isometry type in previously inaccessible cases.

Quasi-isometric invariants remain the foundational machinery enabling large-scale structural classification, geometric rigidity, and the transfer of local-to-global geometric phenomena in geometric group theory and beyond.