a hyperbolic 3-manifold

Hyperbolic geometry is the geometry of constant negative curvature — the world where triangles are thin, parallel lines diverge, and the area of a disc grows exponentially with its radius. A hyperbolic 3-manifold is a three-dimensional space that is locally exactly this geometry everywhere: zoom in anywhere and you see hyperbolic 3-space H^3. Such manifolds are the overwhelmingly typical 3-manifolds, and they are where the deepest rigidity phenomena of low-dimensional geometry live.

Precisely, a complete hyperbolic 3-manifold is a quotient M = H^3 / Gamma, where H^3 is hyperbolic 3-space (constant curvature -1) and Gamma is a discrete, torsion-free subgroup of its orientation-preserving isometry group PSL(2,C) — a torsion-free Kleinian group acting freely and properly discontinuously. Equivalently it is a manifold carrying a complete (PSL(2,C), H^3)-structure, with holonomy the inclusion of Gamma. Finite-volume examples include closed manifolds and 'cusped' manifolds (like knot complements) whose ends are tubes modeled on a quotient of a horoball, each cusp cross-section a flat torus shrinking off to infinity.

Their structure is governed by the thick-thin decomposition: by the Margulis lemma there is a universal constant such that, below it, the manifold splits into a 'thick part' (injectivity radius bounded below, geometrically tame and compact up to finitely many shapes) and a 'thin part' consisting of tubes around short geodesics and cusp neighborhoods. The headline rigidity is Mostow-Prasad: in dimension ≥3 a finite-volume hyperbolic structure is UNIQUE up to isometry, completely determined by the underlying topology — so quantities like hyperbolic volume become topological invariants. This is precisely the dimension-≥3 phenomenon that FAILS for surfaces, where Teichmüller space is a whole moduli of hyperbolic structures.