Ultralimit

Ultrafilters

Recall that an ultrafilter ω on the set of natural numbers ℕ is a set of subsets of ℕ (whose inclusion function can thought of as a measure) which is closed under finite intersection, upwards-closed, and which, given any subset X of ℕ, contains either X or ℕ∖ X. An ultrafilter ω on ℕ 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 N . If ( x n ) n ∈ N is a sequence of points in a metric space (X,d) and x∈ X, the point x is called the ω -limit of x_n, denoted x = lim ω x n , if for every ϵ > 0 we have:

It is not hard to see the following:

An important basic fact states that, if (X,d) is compact and ω is a non-principal ultrafilter on N , 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 R (as closed intervals are compact).

Ultralimit of metric spaces with specified base-points

Let ω be a non-principal ultrafilter on N . Let (X_n,d_n) be a sequence of metric spaces with specified base-points p_n∈X_n.

Let us say that a sequence ( x n ) n ∈ N , where x_n∈X_n, is admissible, if the sequence of real numbers (d_n(x_n,p_n))_n is bounded, that is, if there exists a positive real number C such that d n ( x n , p n ) ≤ C . Let us denote the set of all admissible sequences by A .

It is easy to see from the triangle inequality that for any two admissible sequences x = ( x n ) n ∈ N and y = ( y n ) n ∈ N the sequence (d_n(x_n,y_n))_n is bounded and hence there exists an ω-limit d ^ ∞ ( x , y ) := lim ω d n ( x n , y n ) . Let us define a relation ∼ on the set A of all admissible sequences as follows. For x , y ∈ A we have x ∼ y whenever d ^ ∞ ( x , y ) = 0. It is easy to show that ∼ is an equivalence relation on A .

The ultralimit with respect to ω of the sequence (X_n,d_n, p_n) is a metric space ( X ∞ , d ∞ ) defined as follows.

As a set, we have X ∞ = A / ∼ .

For two ∼ -equivalence classes [ x ] , [ y ] of admissible sequences x = ( x n ) n ∈ N and y = ( y n ) n ∈ N we have d ∞ ( [ x ] , [ y ] ) := d ^ ∞ ( x , y ) = lim ω d n ( x n , y n ) .

It is not hard to see that d ∞ is well-defined and that it is a metric on the set X ∞ .

Denote ( X ∞ , d ∞ ) = lim ω ( X n , d n , p n ) .

On basepoints in the case of uniformly bounded spaces

Suppose that (X_n,d_n) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C>0 such that diam(X_n)≤C for every n ∈ N . Then for any choice p_n of base-points in X_n every sequence ( x n ) n , x n ∈ X n is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit ( X ∞ , d ∞ ) depends only on (X_n,d_n) and on ω but does not depend on the choice of a base-point sequence p n ∈ X n . . In this case one writes ( X ∞ , d ∞ ) = lim ω ( X n , d n ) .

Basic properties of ultralimits

1. If (X_n,d_n) are geodesic metric spaces then ( X ∞ , d ∞ ) = lim ω ( X n , d n , p n ) is also a geodesic metric space.
2. If (X_n,d_n) are complete metric spaces then ( X ∞ , d ∞ ) = lim ω ( X n , d n , p n ) is also a complete metric space.

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

1. If (X_n,d_n) are compact metric spaces that converge to a compact metric space (X,d) in the Gromov–Hausdorff sense (this automatically implies that the spaces (X_n,d_n) have uniformly bounded diameter), then the ultralimit ( X ∞ , d ∞ ) = lim ω ( X n , d n ) is isometric to (X,d).
2. Suppose that (X_n,d_n) are proper metric spaces and that p n ∈ X n are base-points such that the pointed sequence (X_n,d_n,p_n) converges to a proper metric space (X,d) in the Gromov–Hausdorff sense. Then the ultralimit ( X ∞ , d ∞ ) = lim ω ( X n , d n , p n ) is isometric to (X,d).
3. Let κ≤0 and let (X_n,d_n) be a sequence of CAT(κ)-metric spaces. Then the ultralimit ( X ∞ , d ∞ ) = lim ω ( X n , d n , p n ) is also a CAT(κ)-space.
4. Let (X_n,d_n) be a sequence of CAT(κ_n)-metric spaces where lim n → ∞ κ n = − ∞ . Then the ultralimit ( X ∞ , d ∞ ) = lim ω ( X n , d n , p n ) 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 N and let p_n ∈ X be a sequence of base-points. Then the ω–ultralimit of the sequence ( X , d n , p n ) is called the asymptotic cone of X with respect to ω and ( p n ) n and is denoted C o n e ω ( X , d , ( p n ) n ) . One often takes the base-point sequence to be constant, p_n = p for some p ∈ X; in this case the asymptotic cone does not depend on the choice of p ∈ X and is denoted by C o n e ω ( X , d ) or just C o n e ω ( 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 (X_n,d_n)=(X,d) for every n ∈ N . Then the ultralimit ( X ∞ , d ∞ ) = lim ω ( X n , d n ) is isometric to (X,d).
2. Let (X,d_X) and (Y,d_Y) be two distinct compact metric spaces and let (X_n,d_n) be a sequence of metric spaces such that for each n either (X_n,d_n)=(X,d_X) or (X_n,d_n)=(Y,d_Y). Let A 1 = { n | ( X n , d n ) = ( X , d X ) } and A 2 = { n | ( X n , d n ) = ( Y , d Y ) } . Thus A₁, A₂ are disjoint and A 1 ∪ A 2 = N . Therefore, one of A₁, A₂ has ω-measure 1 and the other has ω-measure 0. Hence lim ω ( X n , d n ) is isometric to (X,d_X) if ω(A₁)=1 and lim ω ( X n , d n ) is isometric to (Y,d_Y) 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 basepoint p∈M. Then the ultralimit (and even the ordinary Gromov-Hausdorff limit) lim ω ( M , n d , p ) is isometric to the tangent space T_pM of M at p with the distance function on T_pM given by the inner product g(p). Therefore, the ultralimit lim ω ( M , n d , p ) is isometric to the Euclidean space R m with the standard Euclidean metric.
4. Let ( R m , d ) be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone C o n e ω ( R m , d ) is isometric to ( R m , d ) .
5. Let ( Z 2 , d ) 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 C o n e ω ( Z 2 , d ) is isometric to ( R 2 , d 1 ) where d 1 is the Taxicab metric (or L¹-metric) on R 2 .
6. Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone C o n e ω ( X ) is a real tree.
7. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone C o n e ω ( X ) is a single point.
8. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone C o n e ω ( X ) is also a CAT(0)-space.