Editing Finitely generated group (section)

=== Differential geometry and topology ===

* [[Fundamental group|Fundamental groups]] of compact [[Manifold|manifolds]] are finitely generated. Their geometry coarsely reflects the possible geometries of the manifold: for instance, non-positively curved compact manifolds have [[CAT(0) group|CAT(0)]] fundamental groups, whereas uniformly positively-curved manifolds have finite fundamental group (see [[Myers's theorem#Corollaries|Myers' theorem]]).
* [[Mostow rigidity theorem|Mostow's rigidity theorem]]: for compact [[Hyperbolic manifold|hyperbolic manifolds]] of dimension at least 3, an isomorphism between their fundamental groups extends to a [[Isometry (Riemannian geometry)|Riemannian isometry]].
* [[Mapping class group of a surface|Mapping class groups of surfaces]] are also important finitely generated groups in low-dimensional topology.