In
geometric group theory, the Rips machine is a method of studying the
action of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
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 It was introduced in unpublished work of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
that any
finitely generated group acting freely on an R-tree is a
free product of free abelian and surface groups.
Actions of surface groups on R-trees
By
Bass–Serre theory, a group acting freely on a simplicial tree is free. This is no longer true for R-trees, as Morgan and Shalen showed that the fundamental groups of surfaces of
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
less than −1 also act freely on a R-trees.
They proved that the fundamental group of a connected closed surface S acts freely on an R-tree if and only if S is not one of the 3 nonorientable surfaces of Euler characteristic ≥−1.
Applications
The Rips machine assigns to a stable isometric action of a finitely generated group ''G'' a certain "normal form" approximation of that action by a stable action of ''G'' on a simplicial tree and hence a splitting of ''G'' in the sense of Bass–Serre theory. Group actions on
real trees arise naturally in several contexts in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
: for example as boundary points of the
Teichmüller space (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
-tree endowed with an isometric action of the fundamental group of the surface), as
Gromov-Hausdorff limits of, appropriately rescaled,
Kleinian group actions,
and so on. The use of
-trees machinery provides substantial shortcuts in modern proofs of
Thurston's Hyperbolization Theorem for
Haken 3-manifolds.
Similarly,
-trees play a key role in the study of
Culler-
Vogtmann
Karen Vogtmann (born July 13, 1949 in Pittsburg, California[''Biographies of Candidates 20 ...](_blank)
's Outer space as well as in other areas of
geometric group theory; for example,
asymptotic cones of groups often have a tree-like structure and give rise to group actions on
real trees. The use of
-trees, together with Bass–Serre theory, is a key tool in the work of Sela on solving the isomorphism problem for (torsion-free)
word-hyperbolic groups, Sela's version of the JSJ-decomposition theory and the work of Sela on the Tarski Conjecture for free groups and the theory of
limit groups.
References
Further reading
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{{DEFAULTSORT:Rips Machine
Hyperbolic geometry
Geometric group theory
Trees (topology)