Mostow

Mostow is a surname most notably associated with the mathematician G. D. Mostow, who proved a rigidity theorem in hyperbolic geometry in 1968. The result, now known as Mostow rigidity, has had a profound influence on geometric topology and the theory of lattices in Lie groups.

Mostow's rigidity theorem states that if M and N are complete, finite-volume hyperbolic n-manifolds with dimension n at least 3, and there is an isomorphism between their fundamental groups π1(M) ≅ π1(N), then the isomorphism is induced by a unique isometry between M and N. Consequently, the hyperbolic structure of such a manifold is uniquely determined by its fundamental group; there are no nontrivial deformations of the hyperbolic metric within this setting. This rigidity implies that topological data constrain the geometry more tightly than in lower dimensions.

The theorem was extended to finite-volume manifolds by Grigory Prasad, yielding the combined result often referred to as Mostow–Prasad rigidity. The rigidity phenomenon plays a central role in the study of lattices in Lie groups and in geometric topology, with consequences including the invariance of volume under isomorphisms of fundamental groups and strong constraints on possible geometric structures on manifolds.

In contrast, Mostow rigidity does not hold in dimension two; hyperbolic structures on surfaces exhibit moduli spaces of deformations described by Teichmüller theory. Thus Mostow rigidity is a feature unique to dimensions three and higher.