# Ultralimit

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces $X_{n}$. The concept captures the limiting behavior of finite configurations in the $X_{n}$ spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.

## Ultrafilters

An ultrafilter, denoted as ω, on the set of natural numbers $\mathbb {N}$ is a set of nonempty subsets of $\mathbb {N}$ (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 $\mathbb {N}$, contains either X or $\mathbb {N}$\ X. An ultrafilter on $\mathbb {N}$ 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 $\mathbb {N}$.
If $(x_{n})_{n\in \mathbb {N} }$ is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called ω-limit of x$_n$, denoted as $x=\lim _{\omega }x_{n}$, if for every $\epsilon >0$ it holds that
    $\{n:d(x_{n},x)\leq \epsilon \}\in \omega .$
It is observed that,
* 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 should be non-principal.)
A fundamental fact states that, if (X,d) is compact and ω is a non-principal Ultrafilter on $\mathbb {N}$, 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 $\mathbb {R}$, 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$_n$.
Suppose 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 $d_{n}(x_{n},p_{n})\leq C$, then denote the set of all admissible sequences by ${\mathcal {A}}$.
It follows 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 ${\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} ):=\lim _{\omega }d_{n}(x_{n},y_{n})$. One can define a relation $\sim$ on the set ${\mathcal {A}}$ of all admissible sequences as follows. For $\mathbf {x} ,\mathbf {y} \in {\mathcal {A}}$, there is $\mathbf {x} \sim \mathbf {y}$ whenever ${\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} )=0.$ This helps 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.
Written as a set, $X_{\infty }={\mathcal {A}}/{\sim }$ .
For two $\sim$-equivalence classes $[\mathbf {x} ],[\mathbf {y} ]$ of admissible sequences $\mathbf {x} =(x_{n})_{n\in \mathbb {N} }$ and $\mathbf {y} =(y_{n})_{n\in \mathbb {N} }$, there is $d_{\infty }([\mathbf {x} ],[\mathbf {y} ]):={\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} )=\lim _{\omega }d_{n}(x_{n},y_{n}).$
This shows 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 base points 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 also a geodesic metric space.
* If (X$_n$,d$_n$) are complete metric spaces then $(X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})$ 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.
* 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).
* 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.
* 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.

## 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. 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 (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 $A_{2}=\{n|(X_{n},d_{n})=(Y,d_{Y})\}\,$. Thus A$_1$, A$_2$ are disjoint and $A_{1}\cup A_{2}=\mathbb {N} .$ Therefore, one of A$_1$, A$_2$ 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(A$_1$)=1 and $\lim _{\omega }(X_{n},d_{n})$ is isometric to (Y,d$_Y$) if ω(A$_2$)=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 base point p∈M. Then the ultralimit (and even the ordinary Gromov-Hausdorff limit) $\lim _{\omega }(M,nd,p)$ is isometric to the tangent space T$_p$M of M at p with the distance function on T$_p$M 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.
* 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 L$^1$-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.
* 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.

## Footnotes

* 1 2 3 4 5 6 7 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
* ↑ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; 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 978-0-8218-3332-2; Proposition 7.20, p. 108.
* ↑ Bridson, Haefliger "Metric Spaces of Non-positive curvature" Lemma 5.53
* ↑ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2
* ↑ 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 978-0-8218-3332-2; Example 7.30, p. 118.