Rips machine

In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.

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]

  1. ^ Morgan, John W.; Shalen, Peter B. (1991), "Free actions of surface groups on R-trees", Topology, 30 (2): 143–154, doi:10.1016/0040-9383(91)90002-L, ISSN 0040-9383, MR 1098910
  2. ^ Bestvina, Mladen; Feighn, Mark (1995), "Stable actions of groups on real trees", Inventiones Mathematicae, 121 (2): 287–321, doi:10.1007/BF01884300, ISSN 0020-9910, MR 1346208, S2CID 122048815