Ultralimit

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 $\mathbb {N}$ is a finite-additive set function (which can be thought of as a measure) $\omega :2^{\mathbb {N} }\to \{0,1\}$ from the power set $2^{\mathbb {N} }$ (that is, the set of all subsets of $\mathbb {N}$ ) to the set {0,1} such that $\omega (\mathbb {N} )=1$ . An ultrafilter ω on $\mathbb {N}$ is non-principal if for every finite subset $F\subseteq \mathbb {N}$ we have ω(F)=0.

Limit of a sequence of points with respect to an ultrafilter

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

\omega \{n:d(x_{n},x)\leq \epsilon \}=1.

It is not hard to see the following:

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

An important basic fact^[1] states that, if (X,d) is a compact metric space and ω is a non-principal ultrafilter on $\mathbb {N}$ , 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 $\mathbb {N}$ . 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 $(x_{n})_{n\in \mathbb {N} }$ , 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 $d_{n}(x_{n},p_{n})\leq C$ . 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 $\mathbf {x} =(x_{n})_{n\in \mathbb {N} }$ and $\mathbf {y} =(y_{n})_{n\in \mathbb {N} }$ the sequence (d_n(x_n,y_n))_n is bounded and hence there exists an ω-limit ${\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} ):=\lim _{\omega }d_{n}(x_{n},y_{n})$ . Let us define a relation $\sim$ on the set ${\mathcal {A}}$ the set of all admissible sequences as follows. For $\mathbf {x} ,\mathbf {y} \in {\mathcal {A}}$ we have $\mathbf {x} \sim \mathbf {y}$ whenever ${\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} )=0.$ 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 $(X_{\infty },d_{\infty })$ defined as follows. ^[2]

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

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

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

Denote $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ .

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 $n\in \mathbb {N}$ . 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 $(X_{\infty },d_{\infty })$ 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 $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n})$ .

Basic properties of ultralimits

If (X_n,d_n) are geodesic metric spaces then $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ is a also geodesic metric space.^[1]
If (X_n,d_n) are complete metric spaces then $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ is a also complete metric space.^[3]^[4]
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 $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n})$ is isometric to (X,d).
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 $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ is isometric to (X,d). ^[1]
Let κ≤0 and let (X_n,d_n) be a sequence of CAT(κ)-metric spaces. Then the ultralimit $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ is also a CAT(κ)-space. ^[1]
Let (X_n,d_n) be a sequence of CAT(κ_n)-metric spaces where $\lim _{n\to \infty }\kappa _{n}=-\infty .$ Then the ultralimit $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ 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 $\mathbb {N}$ 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 $Cone_{\omega }(X,d,(p_{n})_{n})\,$ . 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 $Cone_{\omega }(X,d)\,$ or just $Cone_{\omega }(X)\,$ .

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.^[5] Asymptotic cones turned also out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.^[6]

Examples

Let (X,d) be a compact metric space and put (X_n,d_n)=(X,d) for every $n\in \mathbb {N}$ . Then the ultralimit $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n})$ is isometric to (X,d).
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 $A_{2}=\{n|(X_{n},d_{n})=(X,d_{X})\}\,$ . Thus A₁, A₂ are disjoint and $A_{1}\cup A_{2}=\mathbb {N} .$ Therefore one of A₁, A₂ has ω-measure 1 and the other has ω-measure 0. Hence $\lim _{\omega }(X_{n},d_{n})$ is isometric to (X,d_X) if ω(A₁)=1 and $\lim _{\omega }(X_{n},d_{n})$ is isometric to (Y,d_Y) if ω(A₂)=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
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) $\lim _{\omega }(M,nd,p)$ 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 $\lim _{\omega }(M,nd,p)$ is isometric to the Euclidean space $\mathbb {R} ^{m}$ with the standard Euclidean metric.^[7]
Let $(\mathbb {R} ^{m},d)$ be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone $Cone_{\omega }(\mathbb {R} ^{m},d)$ is isometric to $(\mathbb {R} ^{m},d)$ .
Let $(\mathbb {Z} ^{2},d)$ 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 $Cone_{\omega }(\mathbb {Z} ^{2},d)$ is isometric to $(\mathbb {R} ^{2},d_{1})$ where $d_{1}\,$ is the Taxicab metric (or L¹-metric) on $\mathbb {R} ^{2}$ .
Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone $Cone_{\omega }(X)\,$ is a real tree.^[1]^[8]
Let (X,d) be a metric space of finite diameter. Then the asymptotic cone $Cone_{\omega }(X)\,$ is a single point.
Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone $Cone_{\omega }(X)\,$ is also a CAT(0)-space.^[1]

Footnotes

^ ^a ^b ^c ^d ^e ^f ^g M. Kapovich B. Leeb. On asymptotic cones and quasi-isometry classes of fundamental groups of nonpositively curved manifolds, Geometric and Functional Analysis, Vol. 5 (1995), no. 3, pp. 582–603
^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 9780821833322; Definition 7.19, p. 107.
^ 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.
^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 9780821833322; Proposition 7.20, p. 108.
^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 9780821833322
^ 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.
^ 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
^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 9780821833322; Example 7.30, p. 118.

Basic References

John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 9780821833322; 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 nonpositively curved manifolds, Geometric and Functional Analysis, Vol. 5 (1995), no. 3, pp. 582-603
M. Kapovich. Hyperbolic Manifolds and Discrete Groups. Birkhäuser, 2000. ISBN 9780817639044; 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:0817638989; 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.