Ultralimit


In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces Xn a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces Xn and uses an ultrafilter to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion of Gromov–Hausdorff convergence of metric spaces.

Ultrafilters

Recall that an ultrafilter ω on the set of natural numbers is a set of nonempty subsets of which is closed under finite intersection, upwards-closed, and which, given any subset X of, contains either X or. An ultrafilter ω on is non-principal if it contains no finite set.

Limit of a sequence of points with respect to an ultrafilter

Let ω be a non-principal ultrafilter on.
If is a sequence of points in a metric space and x∈ X, the point x is called the ω -limit of xn, denoted, if for every we have:
It is not hard to see the following:
An important basic fact states that, if is compact and ω is a non-principal ultrafilter on, the ω-limit of any sequence of points in X exists.
In particular, any bounded sequence of real numbers has a well-defined ω-limit in .

Ultralimit of metric spaces with specified base-points

Let ω be a non-principal ultrafilter on. Let be a sequence of metric spaces with specified base-points pn∈Xn.
Let us say that a sequence, where xn∈Xn, is admissible, if the sequence of real numbers n is bounded, that is, if there exists a positive real number C such that .
Let us denote the set of all admissible sequences by.
It is easy to see from the triangle inequality that for any two admissible sequences and the sequence n is bounded and hence there exists an ω-limit . Let us define a relation on the set of all admissible sequences as follows. For we have whenever It is easy to show that is an equivalence relation on
The ultralimit with respect to ω of the sequence is a metric space defined as follows.
As a set, we have .
For two -equivalence classes of admissible sequences and we have
It is not hard to see 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 is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C>0 such that diam≤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 and on ω but does not depend on the choice of a base-point sequence. In this case one writes.

Basic properties of ultralimits

  1. If are geodesic metric spaces then is also a geodesic metric space.
  2. If are complete metric spaces then is also a complete metric space.
Actually, by construction, the limit space is always complete, even when
is a repeating sequence of a space which is not complete.
  1. If are compact metric spaces that converge to a compact metric space in the Gromov–Hausdorff sense, then the ultralimit is isometric to.
  2. Suppose that are proper metric spaces and that are base-points such that the pointed sequence converges to a proper metric space in the Gromov–Hausdorff sense. Then the ultralimit is isometric to.
  3. Let κ≤0 and let be a sequence of CAT-metric spaces. Then the ultralimit is also a CAT-space.
  4. Let be a sequence of CAT-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 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 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 be a compact metric space and put = for every. Then the ultralimit is isometric to.
  2. Let and be two distinct compact metric spaces and let be a sequence of metric spaces such that for each n either = or =. 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 if ω=1 and is isometric to if ω=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
  3. Let 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 is a geodesic metric space. Choose a basepoint p∈M. Then the ultralimit is isometric to the tangent space TpM of M at p with the distance function on TpM given by the inner product g. Therefore, the ultralimit is isometric to the Euclidean space with the standard Euclidean metric.
  4. Let be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone is isometric to.
  5. 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 on .
  6. Let be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone is a real tree.
  7. Let be a metric space of finite diameter. Then the asymptotic cone is a single point.
  8. Let be a CAT-metric space. Then the asymptotic cone is also a CAT-space.

    Footnotes