nLab Mostow rigidity theorem

Statement

If a 3-manifold $\Sigma$ (or more generally a smooth manifold of dimension $\geq 3$) admits the structure of a hyperbolic manifold of finite volume, then this structure is unique up to isometry, and in fact it is uniquely determined already by the fundamental group $\pi_1(\Sigma)$.

(Mostow 68).

References

The original article:

• George Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 53-104 (numdam:PMIHES_1968__34__53_0)

Review:

• Marc Bourdon, Mostow type rigidity theorems, In: Handbook of Group Actions (Vol. IV), Advanced Lectures in Mathematics 41, Ch. 4, pp. 139–188, International Press 2018 (pdf, book toc pdf)

See also

• Wikipedia, Mostow rigidity theorem