Mikhail Gromov

Mikhail Leonidovich Gromov (born 1943) is a Franco-Russian mathematician whose work spans geometric group theory, Riemannian geometry, symplectic geometry, and metric geometry. Born in Boksitogorsk, Russia, Gromov studied at Leningrad State University before emigrating to the United States in 1974 and later settling in France, where he holds a permanent position at the Institut des Hautes Etudes Scientifiques (IHES).

Gromov’s approach to mathematics is characterized by the introduction of broad, flexible frameworks that reorganize entire fields. His concept of hyperbolic groups (introduced in a 1987 monograph) redefined geometric group theory by identifying a large and natural class of groups whose algebraic properties can be studied through the geometry of their Cayley graphs. A group is hyperbolic in Gromov’s sense if its Cayley graph satisfies a thin-triangle condition — a metric property that captures, in a combinatorial setting, the negative curvature of hyperbolic geometry.

His work on quasi-isometric embeddings established that many geometric properties of metric spaces are invariant under maps that distort distances by bounded multiplicative and additive factors. This perspective — studying spaces up to quasi-isometry rather than isometry — has become a standard lens in geometric group theory and coarse geometry.

In Riemannian geometry, Gromov introduced the concept of Gromov-Hausdorff distance, which provides a way to measure how far apart two metric spaces are from being isometric. This tool made it possible to speak rigorously about limits and convergence of sequences of Riemannian manifolds, opening the door to compactness theorems in geometry.

Gromov received the Abel Prize in 2009. His influence extends beyond pure mathematics: his geometric framework for metric spaces has been applied to the study of perceptual spaces in olfactory research, where the distances between odor representations can be analyzed using tools from metric geometry.

Related terms

• metric space — the mathematical structure Gromov’s work generalizes
• geometric group theory — the field Gromov’s hyperbolic group theory helped establish