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Theory of hyperbolic groups and asymptotic geometry — Mikhail Gromov

1987 AD · Transmission: Global
MathematicsTheoryRussian

Throughout the 1980s, Mikhail Gromov profoundly transforms how mathematicians study infinite groups, introducing a radically new geometric perspective: instead of analyzing a group through its internal algebraic properties, Gromov proposes treating it as a geometric object in its own right, equipping it with a metric — the so-called word metric — and studying its large-scale geometry, deliberately ignoring fine short-range details. His 1987 paper, "Hyperbolic Groups", introduces and formalizes the class of hyperbolic groups: those whose large-scale geometry resembles that of hyperbolic space, a concept that enormously generalizes the more restricted notion William Thurston had developed for the specific case of three-dimensional hyperbolic manifolds. Gromov also develops Gromov-Hausdorff convergence, a tool allowing rigorous comparison of metric spaces of very different natures — comparing, for example, how a curved space looks "from very far away" — which would prove decisive for the later development of Ricci geometry and, notably, for the tools Grigori Perelman would employ two decades later in his proof of the Poincaré conjecture. His body of work — which also includes foundational contributions to symplectic geometry via pseudoholomorphic curves and Gromov-Witten invariants, and to the study of partial differential equations via the h-principle — has been described as a revolution of Riemannian geometry as a whole, and established geometric group theory as its own distinct mathematical discipline.

InstitutionInstitut des Hautes Études Scientifiques (IHÉS)
Historical regionFrance
Primary sourceGromov, M. — "Hyperbolic Groups" in Essays in Group Theory, Mathematical Sciences Research Institute Publications vol. 8 (Springer, 1987): 75-263
Secondary sourceAbel Prize 2009 — Citation and Scientific Background (abelprize.no)
Original languageEnglish
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