Ultralimit

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces [math]\displaystyle{ X_n }[/math]. The concept captures the limiting behavior of finite configurations in the [math]\displaystyle{ X_n }[/math] spaces employing an ultrafilter to bypass the need for repeatedly consideration of subsequences to ensure convergence. Ultralimits generalize Gromov Hausdorff convergence in metric spaces.

## Contents

* 1 Ultrafilters
* 2 Limit of a sequence of points with respect to an Ultrafilter
* 3 Ultralimit of metric spaces with specified base-points
* 4 On base points in the case of uniformly bounded spaces
* 5 Basic properties of Ultralimits
* 6 Asymptotic cones
* 7 Examples
* 8 Footnotes
* 9 References
* 10 See also

## Ultrafilters

An Ultrafilter, denoted as ω, on the set of natural numbers [math]\displaystyle{ \mathbb{N} }[/math] is a set of nonempty subsets of [math]\displaystyle{ \mathbb{N} }[/math] (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset X of [math]\displaystyle{ \mathbb{N} }[/math], contains either X or [math]\displaystyle{ \mathbb{N} }[/math]\ X. An Ultrafilter on [math]\displaystyle{ \mathbb{N} }[/math] is non-principal if it contains no finite set.

## Limit of a sequence of points with respect to an Ultrafilter

In the following, ω is a non-principal Ultrafilter on [math]\displaystyle{ \mathbb N }[/math].

If [math]\displaystyle{ (x_n)_{n\in \mathbb N} }[/math] is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called ω-limit of xn, denoted as [math]\displaystyle{ x=\lim_\omega x_n }[/math], if for every [math]\displaystyle{ \epsilon\gt 0 }[/math] it holds that

[math]\displaystyle{ \\{n: d(x_n,x)\le \epsilon \\}\in\omega. }[/math]

It is observed that,

* If an ω-limit of a sequence of points exists, it is unique.
* If [math]\displaystyle{ x=\lim_{n\to\infty} x_n }[/math] in the standard sense, [math]\displaystyle{ x=\lim_\omega x_n }[/math]. (For this property to hold, it is crucial that the Ultrafilter should be non-principal.)

A fundamental fact[1] states that, if (X,d) is compact and ω is a non-principal Ultrafilter on [math]\displaystyle{ \mathbb N }[/math], the ω-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 ω-limit in [math]\displaystyle{ \mathbb R }[/math], as closed intervals are compact.

## Ultralimit of metric spaces with specified base-points

Let ω be a non-principal Ultrafilter on [math]\displaystyle{ \mathbb N }[/math]. Let (Xn ,dn) be a sequence of metric spaces with specified base-points pn ∈ Xn.

Suppose that a sequence [math]\displaystyle{ (x_n)_{n\in\mathbb N} }[/math], 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 such that [math]\displaystyle{ d_n(x_n,p_n)\le C }[/math], then denote the set of all admissible sequences by [math]\displaystyle{ \mathcal A }[/math].

It follows from the triangle inequality that for any two admissible sequences [math]\displaystyle{ \mathbf x=(x_n)_{n\in\mathbb N} }[/math] and [math]\displaystyle{ \mathbf y=(y_n)_{n\in\mathbb N} }[/math] the sequence (dn(xn,yn))n is bounded and hence there exists an ω-limit [math]\displaystyle{ \hat d_\infty(\mathbf x, \mathbf y):=\lim_\omega d_n(x_n,y_n) }[/math]. One can define a relation [math]\displaystyle{ \sim }[/math] on the set [math]\displaystyle{ \mathcal A }[/math] of all admissible sequences as follows. For [math]\displaystyle{ \mathbf x, \mathbf y\in \mathcal A }[/math], there is [math]\displaystyle{ \mathbf x\sim\mathbf y }[/math] whenever [math]\displaystyle{ \hat d_\infty(\mathbf x, \mathbf y)=0. }[/math] This helps to show that [math]\displaystyle{ \sim }[/math] is an equivalence relation on [math]\displaystyle{ \mathcal A. }[/math]

The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space [math]\displaystyle{ (X_\infty, d_\infty) }[/math] defined as follows.[2]

Written as a set, [math]\displaystyle{ X_\infty=\mathcal A/{\sim} }[/math] .

For two [math]\displaystyle{ \sim }[/math]-equivalence classes [math]\displaystyle{ [\mathbf x], [\mathbf y] }[/math] of admissible sequences [math]\displaystyle{ \mathbf x=(x_n)_{n\in\mathbb N} }[/math] and [math]\displaystyle{ \mathbf y=(y_n)_{n\in\mathbb N} }[/math], there is [math]\displaystyle{ d_\infty([\mathbf x], [\mathbf y]):=\hat d_\infty(\mathbf x,\mathbf y)=\lim_\omega d_n(x_n,y_n). }[/math]

This shows that [math]\displaystyle{ d_\infty }[/math] is well-defined and that it is a metric on the set [math]\displaystyle{ X_\infty }[/math].

Denote [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n, p_n) }[/math] .

## 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 for every [math]\displaystyle{ n\in \mathbb N }[/math]. Then for any choice pn of base-points in Xn every sequence [math]\displaystyle{ (x_n)_n, x_n\in X_n }[/math] is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit [math]\displaystyle{ (X_\infty, d_\infty) }[/math] depends only on (Xn,dn) and on ω but does not depend on the choice of a base-point sequence [math]\displaystyle{ p_n\in X_n }[/math]. In this case one writes [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n) }[/math].

## Basic properties of Ultralimits

1. If (Xn,dn) are geodesic metric spaces then [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n, d_n, p_n) }[/math] is also a geodesic metric space.[1]
2. If (Xn,dn) are complete metric spaces then [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n, p_n) }[/math] is also a complete metric space.[3][4]

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.[5]

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) have uniformly bounded diameter), then the ultralimit [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n) }[/math] is isometric to (X,d).
2. Suppose that (Xn,dn) are proper metric spaces and that [math]\displaystyle{ p_n\in X_n }[/math] are base-points such that the pointed sequence (Xn,dn,pn) converges to a proper metric space (X,d) in the Gromov–Hausdorff sense. Then the ultralimit [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n,p_n) }[/math] is isometric to (X,d).[1]
3. Let κ≤0 and let (Xn,dn) be a sequence of CAT(κ)-metric spaces. Then the ultralimit [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n, p_n) }[/math] is also a CAT(κ)-space.[1]
4. Let (Xn,dn) be a sequence of CAT(κn)-metric spaces where [math]\displaystyle{ \lim_{n\to\infty}\kappa_n=-\infty. }[/math] Then the ultralimit [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n, p_n) }[/math] is real tree.[1]

## Asymptotic cones

An important class of Ultralimits are the so-called asymptotic cones of metric spaces. Let (X,d) be a metric space, let ω be a non-principal Ultrafilter on [math]\displaystyle{ \mathbb N }[/math] and let pn ∈ X be a sequence of base-points. Then the ω-ultralimit of the sequence [math]\displaystyle{ (X, \frac{d}{n}, p_n) }[/math] is called the asymptotic cone of X with respect to ω and [math]\displaystyle{ (p_n)_n\, }[/math] and is denoted [math]\displaystyle{ Cone_\omega(X,d, (p_n)_n)\, }[/math]. 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 and is denoted by [math]\displaystyle{ Cone_\omega(X,d)\, }[/math] or just [math]\displaystyle{ Cone_\omega(X)\, }[/math].

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.[6] Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.[7]

## Examples

1. Let (X,d) be a compact metric space and put (Xn,dn)=(X,d) for every [math]\displaystyle{ n\in \mathbb N }[/math]. Then the ultralimit [math]\displaystyle{ (X_\infty, d_\infty)=\lim_\omega(X_n,d_n) }[/math] 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). Let [math]\displaystyle{ A_1=\\{n | (X_n,d_n)=(X,d_X)\\}\, }[/math] and [math]\displaystyle{ A_2=\\{n | (X_n,d_n)=(Y,d_Y)\\}\, }[/math]. Thus A1, A2 are disjoint and [math]\displaystyle{ A_1\cup A_2=\mathbb N. }[/math] Therefore, one of A1, A2 has ω-measure 1 and the other has ω-measure 0. Hence [math]\displaystyle{ \lim_\omega(X_n,d_n) }[/math] is isometric to (X,dX) if ω(A1)=1 and [math]\displaystyle{ \lim_\omega(X_n,d_n) }[/math] is isometric to (Y,dY) if ω(A2)=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
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) [math]\displaystyle{ \lim_\omega(M,n d, p) }[/math] is isometric to the tangent space TpM of M at p with the distance function on TpM given by the inner product g(p). Therefore, the ultralimit [math]\displaystyle{ \lim_\omega(M,n d, p) }[/math] is isometric to the Euclidean space [math]\displaystyle{ \mathbb R^m }[/math] with the standard Euclidean metric.[8]
4. Let [math]\displaystyle{ (\mathbb R^m, d) }[/math] be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone [math]\displaystyle{ Cone_\omega(\mathbb R^m, d) }[/math] is isometric to [math]\displaystyle{ (\mathbb R^m, d) }[/math].
5. Let [math]\displaystyle{ (\mathbb Z^2,d) }[/math] be the 2-dimensional integer lattice where the distance between two lattice points is given by the length of the shortest edge-path between them in the grid. Then the asymptotic cone [math]\displaystyle{ Cone_\omega(\mathbb Z^2, d) }[/math] is isometric to [math]\displaystyle{ (\mathbb R^2, d_1) }[/math] where [math]\displaystyle{ d_1\, }[/math] is the Taxicab metric (or L1-metric) on [math]\displaystyle{ \mathbb R^2 }[/math].
6. Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone [math]\displaystyle{ Cone_\omega(X)\, }[/math] is a real tree.[1][9]
7. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone [math]\displaystyle{ Cone_\omega(X)\, }[/math] is a single point.
8. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone [math]\displaystyle{ Cone_\omega(X)\, }[/math] is also a CAT(0)-space.[1]

## Footnotes

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 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
2. ↑ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; Definition 7.19, p. 107.
3. ↑ 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.
4. ↑ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; Proposition 7.20, p. 108.
5. ↑ Bridson, Haefliger "Metric Spaces of Non-positive curvature" Lemma 5.53
6. ↑ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2
7. ↑ 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.
8. ↑ Yu. Burago, M. Gromov, and G. Perel'man. A. D. Aleksandrov spaces with curvatures bounded below (in Russian), Uspekhi Matematicheskih Nauk vol. 47 (1992), pp. 3-51; translated in: Russian Math. Surveys vol. 47, no. 2 (1992), pp. 1-58
9. ↑ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; Example 7.30, p. 118.

## References

* John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; 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. ISBN 978-0-8176-3904-4; 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. ISBN 0-8176-3898-9; 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

* Ultrafilter
* Geometric group theory
* Gromov-Hausdorff convergence

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Ultralimit. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Ultralimit&oldid=3378853"