Rips machine
Encyclopedia
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 It was introduced in unpublished work of Eliyahu Rips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 that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups .
, a group acting freely on a simplicial tree is free. This is no longer true for R-trees, as showed that the fundamental groups of surfaces of Euler characteristic
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.
s arise naturally in several contexts in geometric topology
: 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
'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 tree
s. 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.
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, the Rips machine is a method of studying the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
on R-trees
Real tree
A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
. It was introduced in unpublished work of Eliyahu Rips
Eliyahu Rips
Eliyahu Rips, also Ilya Rips is a Latvian-born Israeli mathematician known for his research in geometric group theory. He became known to the general public following his coauthoring a paper on the Torah Code....
in about 1991.
An R-tree is a uniquely arcwise-connected metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
in which every arc is isometric to some real interval. Rips proved the conjecture of 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 theoryBass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
, a group acting freely on a simplicial tree is free. This is no longer true for R-trees, as showed that the fundamental groups of surfaces of Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
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 treeReal tree
A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
s arise naturally in several contexts in geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
: for example as boundary points of the Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
(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 limitsGromov-Hausdorff convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.-Gromov–Hausdorff distance:...
of, appropriately rescaled, Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
actions, and so on. The use of
-trees machinery provides substantial shortcuts in modern proofs of Thurston's Hyperbolization TheoremGeometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
for Haken 3-manifolds
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
. Similarly,
-trees play a key role in the study of CullerMarc Culler
Marc Edward Culler is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the...
-Vogtmann
Karen Vogtmann
Karen Vogtmann is a U.S. mathematician working primarily in the area of geometric group theory. She is known for having introduced, in a 1986 paper with Marc Culler, an object now known as the Culler–Vogtmann Outer space...
's Outer space as well as in other areas of geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
; for example, asymptotic cones of groups often have a tree-like structure and give rise to group actions on real tree
Real tree
A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
s. 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.