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| CAT(0) spaces | |
|---|---|
| Name | CAT(0) spaces |
| Field | Metric geometry, Geometric group theory |
| Introduced by | Élie Cartan, Mikhail Gromov, Alexander Grothendieck |
| First published | 20th century |
| Related | Hadamard manifold, Hyperbolic space, Euclidean space |
CAT(0) spaces CAT(0) spaces are complete geodesic metric spaces with nonpositive curvature in the sense of comparison geometry. They generalize Hadamard manifolds, relate to Gromov hyperbolic spaces, and play a central role in Geometric group theory, Riemannian geometry, and the study of discrete groups such as Coxeter groups and Right-angled Artin groups.
A CAT(0) space is a geodesic metric space satisfying the Cartan–Alexandrov–Toponogov triangle comparison condition relative to Euclidean plane. The definition uses comparison triangles in Euclidean plane to require that distances across geodesic triangles are no greater than in the corresponding triangle in Euclidean plane. Fundamental properties include uniqueness of geodesics between points (visibility of geodesic segments), contractibility (as in Hadamard manifolds and Eilenberg–MacLane spaces), and the existence of nearest-point projections onto closed convex subsets, paralleling projection properties in Hilbert space and Banach space theory. Important concepts associated with CAT(0) spaces include convexity structures akin to those in Symmetric spaces and fixed-point theorems reminiscent of results for Compact group actions and Amenable group actions.
Standard examples include Euclidean spaces, trees such as those arising from Bass–Serre theory, Real hyperbolic spaces of constant negative curvature, and complete simply connected Riemannian manifolds of nonpositive sectional curvature like Hadamard manifolds. Polyhedral examples include CAT(0) Cube complexes which generalize CAT(0) cube complex constructions used by researchers such as Mladen Bestvina and Michah Sageev. Buildings such as Bruhat–Tits buildings for p-adic numbers and symmetric spaces of noncompact type (e.g. associated with SL(n,R)) provide further examples. Non-examples include spaces with positive curvature such as the standard Spheres, many singular spaces with branching geodesics that violate uniqueness (certain pathological Alexandrov spaces), and many fractal metric spaces like the Sierpiński carpet with its typical geodesic failure.
Geodesics in CAT(0) spaces are unique between any two points, analogous to geodesics in Hadamard manifolds and straight lines in Euclidean space. Convexity notions mirror those in Hilbert space: metric balls and distance functions are convex, and the midpoint operation is well-behaved as in affine structures on Symmetric spaces. The projection onto closed convex subsets is uniquely defined, yielding fixed-point results for isometric actions of compact groups such as Orthogonal groups. Geodesic stability properties connect to boundary behavior studied in Morse theory analogs and to contracting properties used in the analysis of Mapping class group actions.
Groups acting properly discontinuously and cocompactly by isometries on CAT(0) spaces are called CAT(0) groups, a class that intersects with Word-hyperbolic groups, Coxeter groups, Artin groups, and many lattice subgroups of Lie groups. Actions on CAT(0) cube complexes yield powerful structural consequences via the work of Sageev, Haglund, and Wise, linking to separability properties studied by Daniel Wise and applications to 3-manifold theory such as results related to Thurston and the Geometrization conjecture. Fixed-point theorems for group actions generalize classical theorems like those of Bruhat and Tits and inform rigidity phenomena for lattices in Semisimple Lie groups and arithmetic groups such as SL(n,Z).
The boundary at infinity (visual boundary) of a proper CAT(0) space encodes asymptotic geometry and generalizes the sphere at infinity for Hyperbolic spaces and Symmetric spaces. Topological and dynamical structures on the boundary relate to convergence actions studied by researchers including Dennis Sullivan and William Thurston, and to the Patterson–Sullivan measure constructions used in Ergodic theory and counting problems for Kleinian groups. For CAT(0) cube complexes the Roller boundary and the visual boundary provide complementary perspectives, linking to boundary theory used in classification of groups like Mapping class group and in rigidity results for Anosov representations.
CAT(0) conditions imply strong topological consequences such as contractibility (compare to Eilenberg–MacLane space properties) and restrictions on homotopy type analogous to those for Nonpositively curved manifolds. Splitting theorems parallel the de Rham decomposition theorem for Riemannian manifolds, and structure theorems constrain the existence of embedded flats by comparison with Euclidean buildings and Flat torus theorem statements for abelian subgroups. The geometry of CAT(0) spaces influences group theoretic invariants like cohomological dimension, and interacts with algorithmic properties studied in decision problem research for groups such as the Word problem.
Key results include the Flat Torus Theorem, the Splitting Theorem, and the Cartan–Hadamard theorem for nonpositive curvature, with rigidity phenomena exemplified by Mostow-type rigidity for rank-one symmetric spaces and superrigidity theorems for lattices in Semisimple Lie groups proved by Gregory Margulis and extended in contexts involving CAT(0) spaces. Work by Gromov on hyperbolic groups, by Ballmann on nonpositive curvature, and by Bridson and Haefliger on metric spaces of nonpositive curvature provides foundational theorems used in modern rigidity, quasi-isometry classification, and quasi-convexity studies. Recent breakthroughs linking cube complex actions to separability and virtual specialness by Wise, Agol, and Haglund produced major applications to the Virtual Haken conjecture and the understanding of 3-manifold groups.
Category:Metric geometry