An R-tree is a uniquely arcwise-connected metric space in which every arc is isometric to some real interval.
Rips proved the conjecture of Morgan and Shalen[1] that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups.
[2] By Bass–Serre theory, a group acting freely on a simplicial tree is free.
Group actions on real trees arise naturally in several contexts in geometric topology: for example as boundary points of the Teichmüller space[3] (every point in the Thurston boundary of the Teichmüller space is represented by a measured geodesic lamination on the surface; this lamination lifts to the universal cover of the surface and a naturally dual object to that lift is an
-trees machinery provides substantial shortcuts in modern proofs of Thurston's hyperbolization theorem for Haken 3-manifolds.