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For the direct limit of a sequence of ultrapowers, see Ultraproduct.

In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces X_n a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces X_n 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 $\mathbb{\mathbb{N}}$ is a finite-additive set function (which can be thought of as a measure) $\omega :2^{\mathbb{\mathbb{N}}}\to \{0,1\}$ from the power set $2^{\mathbb{\mathbb{N}}}$ (that is, the set of all subsets of $\mathbb{\mathbb{N}}$) to the set {0,1} such that $\omega (\mathbb{\mathbb{N}})=1$. An ultrafilter ω on $\mathbb{\mathbb{N}}$ is non-principal if for every finite subset $F\subseteq \mathbb{\mathbb{N}}$ we have ω(F)=0.

Limit of a sequence of points with respect to an ultrafilter

Let ω be a non-principal ultrafilter on $\mathbb{\mathbb{N}}$. If $(x_n{)}_{n\in \mathbb{\mathbb{N}}}$ 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 $x=\underset{\omega }{\lim }{x}_n$, if for every $ϵ>0$ we have:

$\omega \{n:d(x_n,x)\le ϵ\}=1.$

It is not hard to see the following:

• If an ω -limit of a sequence of points exists, it is unique.
• If $x=\underset{n\to \mathrm{\infty }}{\lim }{x}_n$ in the standard sense, $x=\underset{\omega }{\lim }{x}_n$. (For this property to hold it is crucial that the ultrafilter be non-principal.)

An important basic fact states that, if (X,d) is compact and ω is a non-principal ultrafilter on $\mathbb{\mathbb{N}}$, the ω-limit of any sequence of points in X exists (and necessarily unique).

In particular, any bounded sequence of real numbers has a well-defined ω-limit in $\mathbb{ℝ}$ (as closed intervals are compact).

Ultralimit of metric spaces with specified base-points

Let ω be a non-principal ultrafilter on $\mathbb{\mathbb{N}}$. 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 $(x_n{)}_{n\in \mathbb{\mathbb{N}}}$, 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 $d_n(x_n,p_n)\le C$. Let us denote the set of all admissible sequences by $\mathcal{𝒜}$.

It is easy to see from the triangle inequality that for any two admissible sequences $\mathbf{x}=(x_n{)}_{n\in \mathbb{\mathbb{N}}}$ and $\mathbf{y}=(y_n{)}_{n\in \mathbb{\mathbb{N}}}$ the sequence (d_n(x_n,y_n))_n is bounded and hence there exists an ω-limit ${\hat{d}}_{\mathrm{\infty }}(\mathbf{x},\mathbf{y}):=\underset{\omega }{\lim }{d}_n(x_n,y_n)$. Let us define a relation $\sim$ on the set $\mathcal{𝒜}$ of all admissible sequences as follows. For $\mathbf{x},\mathbf{y}\in \mathcal{𝒜}$ we have $\mathbf{x}\sim \mathbf{y}$ whenever ${\hat{d}}_{\mathrm{\infty }}(\mathbf{x},\mathbf{y})=0.$ It is easy to show that $\sim$ is an equivalence relation on $\mathcal{𝒜}.$

The ultralimit with respect to ω of the sequence (X_n,d_n, p_n) is a metric space $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})$ defined as follows.

As a set, we have $X_{\mathrm{\infty }}=\mathcal{𝒜}/\sim$ .

For two $\sim$-equivalence classes $[\mathbf{x}],[\mathbf{y}]$ of admissible sequences $\mathbf{x}=(x_n{)}_{n\in \mathbb{\mathbb{N}}}$ and $\mathbf{y}=(y_n{)}_{n\in \mathbb{\mathbb{N}}}$ we have $d_{\mathrm{\infty }}([\mathbf{x}],[\mathbf{y}]):={\hat{d}}_{\mathrm{\infty }}(\mathbf{x},\mathbf{y})=\underset{\omega }{\lim }{d}_n(x_n,y_n).$

It is not hard to see that $d_{\mathrm{\infty }}$ is well-defined and that it is a metric on the set $X_{\mathrm{\infty }}$.

Denote $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n,p_n)$ .

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

Basic properties of ultralimits

1. If (X_n,d_n) are geodesic metric spaces then $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n,p_n)$ is also a geodesic metric space.
2. If (X_n,d_n) are complete metric spaces then $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n,p_n)$ 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 $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n)$ is isometric to (X,d).
2. Suppose that (X_n,d_n) are proper metric spaces and that $p_n\in X_n$ 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 $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n,p_n)$ 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 $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n,p_n)$ is also a CAT(κ)-space.
4. Let (X_n,d_n) be a sequence of CAT(κ_n)-metric spaces where $\underset{n\to \mathrm{\infty }}{\lim }{\kappa }_n=-\mathrm{\infty }.$ Then the ultralimit $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n,p_n)$ 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 $\mathbb{\mathbb{N}}$ and let p_n ∈ X be a sequence of base-points. Then the ω–ultralimit of the sequence $(X,\frac{d}{n},p_n)$ is called the asymptotic cone of X with respect to ω and $(p_n{)}_n$ and is denoted $Con{e}_{\omega }(X,d,(p_n{)}_n)$. 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 $Con{e}_{\omega }(X,d)$ or just $Con{e}_{\omega }(X)$.

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 $n\in \mathbb{\mathbb{N}}$. Then the ultralimit $(X_{\mathrm{\infty }},d_{\mathrm{\infty }})=\underset{\omega }{\lim }(X_n,d_n)$ 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 $A_1\cup A_2=\mathbb{\mathbb{N}}.$ Therefore one of A₁, A₂ has ω-measure 1 and the other has ω-measure 0. Hence $\underset{\omega }{\lim }(X_n,d_n)$ is isometric to (X,d_X) if ω(A₁)=1 and $\underset{\omega }{\lim }(X_n,d_n)$ 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) $\underset{\omega }{\lim }(M,nd,p)$ 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 $\underset{\omega }{\lim }(M,nd,p)$ is isometric to the Euclidean space ${\mathbb{ℝ}}^m$ with the standard Euclidean metric.
4. Let $({\mathbb{ℝ}}^m,d)$ be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone $Con{e}_{\omega }({\mathbb{ℝ}}^m,d)$ is isometric to $({\mathbb{ℝ}}^m,d)$.
5. Let $({\mathbb{\mathbb{Z}}}^2,d)$ 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 $Con{e}_{\omega }({\mathbb{\mathbb{Z}}}^2,d)$ is isometric to $({\mathbb{ℝ}}^2,d_1)$ where $d_1$ is the Taxicab metric (or L¹-metric) on ${\mathbb{ℝ}}^2$.
6. Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone $Con{e}_{\omega }(X)$ is a real tree.
7. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone $Con{e}_{\omega }(X)$ is a single point.
8. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone $Con{e}_{\omega }(X)$ is also a CAT(0)-space.

Footnotes

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Basic 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