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Wikipedia / Ultralimit

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.

Ultrafilters

An ultrafilter, denoted as , on the set of natural numbers (whose indicator function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset X, contains either X is non-principal if it contains no finite set.

Limit of a sequence of points with respect to an ultrafilter

In the following, .

is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called an -limit of xn it holds that

It is observed that,

  • If an -limit of a sequence of points exists, it is unique.

  • . (For this property to hold, it is crucial that the ultrafilter should be non-principal.)

A fundamental fact states that, if (X,d) is compact and , the -limit of any sequence of points in X exists (and is necessarily unique). In particular, any bounded sequence of real numbers has a well-defined , as closed intervals are compact.

Ultralimit of metric spaces with specified base-points

Let . Let (Xn ,dn) be a sequence of metric spaces with specified base-points pn ∈ Xn.

, where xn ∈ Xn, is admissible if the sequence of real numbers (dn(xn ,pn))n is bounded, that is, if there exists a positive real number C for all n.

It follows from the triangle inequality the sequence (dn(xn,yn))n is bounded and hence there exists an

The ultralimit with respect to of the sequence (Xn,dn, pn defined as follows.

.

is a metric.

On base points in the case of uniformly bounded spaces

Suppose that (Xn, dn) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C > 0 such that diam(Xn) ≤ C. Then for any choice pn of base-points in Xn every depends only on (Xn, dn) and on .

Basic properties of ultralimits

  1. If (Xn,dn) are geodesic metric spaces is also a geodesic metric space.

  2. If (Xn,dn) are complete metric spaces is also a complete metric space.

Actually, by construction, the limit space is always complete, even when (Xn, dn) is a repeating sequence of a space (X, d) which is not complete.

  1. If (Xn, dn) are compact metric spaces that converge to a compact metric space (X, d) in the Gromov–Hausdorff sense (this automatically implies that the spaces (Xn, dn is isometric to (X, d).

  2. Suppose that (Xn, dn) are proper metric spaces are base-points such that the pointed sequence (Xn, dn, pn) converges to a proper metric space (X, d) in the Gromov–Hausdorff is isometric to (X, d).

  3. Let ≤ 0 and let (Xn, dn) be a sequence of CAT(κ)-metric spaces is also a CAT()-space.

  4. Let (Xn, dn) be a sequence of CAT(κn)-metric spaces is real tree.

Asymptotic cones

An important class of ultralimits are the so-called asymptotic cones of metric spaces. Let (X,d) be a metric space, let and let pn∈X be a sequence of base-points. Then the is called the asymptotic cone of X with respect to . One often takes the base-point sequence to be constant, pn = p for some p ∈ X; in this case the asymptotic cone does not depend on the choice of p ∈ X.

The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular. Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.

Examples

  1. Let (X,d) be a compact metric space and put (Xn,dn)=(X,d is isometric to (X,d).

  2. Let (X,dX) and (Y,dY) be two distinct compact metric spaces and let (Xn,dn) be a sequence of metric spaces such that for each n either (Xn,dn)=(X,dX) or (Xn,dn)=(Y,dY. Thus A1, A2 Therefore, one of A1, A2 has -measure 1 and the other has is isometric to (X,dX) if (A1 is isometric to (Y,dY) if (A2)=1. This shows that the ultralimit can depend on the choice of an ultrafilter .

  3. Let (M,g) be a compact connected Riemannian manifold of dimension m, where g is a Riemannian metric on M. Let d be the metric on M corresponding to g, so that (M,d) is a geodesic metric space. Choose a base point p∈M. Then the ultralimit (and even the ordinary Gromov-Hausdorff limit is isometric to the tangent space TpM of M at p with the distance function on TpM given by the inner product g(p) is isometric to the Euclidean space with the standard Euclidean metric.

  4. be the standard m-dimensional Euclidean space.

  5. be the 2-dimensional integer lattice is the Taxicab metric (or L1.

  6. Let (X,d) be a δ-hyperbolic geodesic metric space for some is a real tree.

  7. Let (X,d is a single point.

  8. Let (X,d) be a CAT(0)-metric space is also a CAT(0)-space.

Footnotes

References

  • John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ; Ch. 7.

  • L.Van den Dries, A.J.Wilkie, On Gromov's theorem concerning groups of polynomial growth and elementary logic. Journal of Algebra, Vol. 89(1984), pp.349–374.

  • M. Kapovich B. Leeb. On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds, Geometric and Functional Analysis, Vol. 5 (1995), no. 3, pp.582–603

  • M. Kapovich. Hyperbolic Manifolds and Discrete Groups. Birkhäuser, 2000. ; Ch. 9.

  • Cornelia Druţu and Mark Sapir (with an Appendix by Denis Osin and Mark Sapir), Tree-graded spaces and asymptotic cones of groups. Topology, Volume 44 (2005), no. 5, pp.959–1058.

  • M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics vol. 152, Birkhäuser, 1999. ; Ch. 3.

  • B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Publications Mathématiques de L'IHÉS. Volume 86, Number 1, December 1997, pp.115–197.

See also