In mathematics, an ultra limit is a geometric construction that assigns a limit metric space to a sequence of metric spaces . The concept of such captures the limiting behavior of finite configurations in the  spaces and employs an ultrafilter to bypass the need for repeatedly considering subsequences to ensure convergence. Ultra limits generalize the idea of Gromov Hausdorff convergence in metric spaces.

Ultrafilters

An ultrafilter, denoted as ω, on the set of natural numbers  is a set of nonempty subsets of  (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 , contains  either X or .  An ultrafilter on  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 .

If  is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called the ω-limit of xn, denoted as .

If for every  there are:

It is observed that,
 If an ω-limit of a sequence of points exists, it is unique.
 If  in the standard sense, . (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 ω is a non-principal ultrafilter on , 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 , as closed intervals are compact.

Ultralimit of metric spaces with specified base-points
Let ω be a non-principal ultrafilter on . Let (Xn ,dn) be a sequence of metric spaces with specified base-points pn ∈ Xn.

Suppose that a sequence , 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  , then denote the set of all admissible sequences by .

It follows from the triangle inequality that for any two admissible sequences  and  the sequence (dn(xn,yn))n is bounded and hence there exists an ω-limit  . One can define a relation  on the set  of all admissible sequences as follows. For , there is  whenever  This helps to show that  is an equivalence relation on 

The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space  defined as follows.

Written as a set,  .

For two -equivalence classes  of admissible sequences  and , there is 

This shows that  is well-defined and that it is a metric on the set  .

Denote  .

On basepoints 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 .  Then for any choice pn of base-points in Xn every sequence  is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit  depends only on (Xn,dn)  and on ω but does not depend on the choice of a base-point sequence . In this case one writes .

Basic properties of ultralimits

If (Xn,dn) are geodesic metric spaces then  is also a geodesic metric space.
If (Xn,dn) are complete metric spaces then  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.
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  is isometric to (X,d).
Suppose that (Xn,dn) are proper metric spaces and that  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  is isometric to (X,d).
Let κ≤0 and let (Xn,dn)  be a sequence of CAT(κ)-metric spaces. Then the ultralimit  is also a CAT(κ)-space.
Let (Xn,dn)  be a sequence of CAT(κn)-metric spaces where  Then the ultralimit  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 ω be a non-principal ultrafilter on  and let pn ∈ X be a sequence of base-points. Then the ω–ultralimit of the sequence  is called the asymptotic cone of X with respect to ω and  and is denoted . 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   or just .

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
Let (X,d) be a compact metric space and put (Xn,dn)=(X,d) for every . Then the ultralimit  is isometric to (X,d).
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  and . Thus A1, A2 are disjoint and   Therefore, one of A1, A2  has ω-measure 1 and the other has ω-measure 0. Hence  is isometric to (X,dX) if ω(A1)=1 and  is isometric to (Y,dY) if ω(A2)=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
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 basepoint 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). Therefore, the ultralimit  is isometric to the Euclidean space  with the standard Euclidean metric.
Let  be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone  is isometric to .
Let  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  is isometric to  where  is the Taxicab metric (or L1-metric) on  .
Let  (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone  is a real tree.
Let (X,d) be a metric space of finite diameter. Then the asymptotic cone  is a single point.
Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone  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
Ultrafilter
Geometric group theory
Gromov-Hausdorff convergence

Geometric group theory
Metric geometry