Ultralimit

Recall that an ultrafilter ω on the set of natural numbers ℕ is a set of nonempty 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.

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

${\displaystyle \{n:d(x_{n},x)\leq \epsilon \}\in \omega .}$

It is not hard to see the following:

• If an ω -limit of a sequence of points exists, it is unique.
• If ${\displaystyle x=\lim _{n\to \infty }x_{n}}$ in the standard sense, ${\displaystyle x=\lim _{\omega }x_{n}}$. (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 compact and ω is a non-principal ultrafilter on ${\displaystyle \mathbb {N} }$, 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 ${\displaystyle \mathbb {R} }$ (as closed intervals are compact).

Let ω be a non-principal ultrafilter on ${\displaystyle \mathbb {N} }$. 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 ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$, 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 ${\displaystyle d_{n}(x_{n},p_{n})\leq C}$. Let us denote the set of all admissible sequences by ${\displaystyle {\mathcal {A}}}$.

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

The ultralimit with respect to ω of the sequence (X_n,d_n, p_n) is a metric space ${\displaystyle (X_{\infty },d_{\infty })}$ defined as follows.[2]

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

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

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

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

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

1. If (X_n,d_n) are geodesic metric spaces then ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})}$ is also a geodesic metric space.[1]
2. If (X_n,d_n) are complete metric spaces then ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})}$ is also a complete metric space.[3][4]

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.[5]

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 ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n})}$ is isometric to (X,d).
2. Suppose that (X_n,d_n) are proper metric spaces and that ${\displaystyle p_{n}\in X_{n}}$ 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 ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})}$ is isometric to (X,d).[1]
3. Let κ≤0 and let (X_n,d_n) be a sequence of CAT(κ)-metric spaces. Then the ultralimit ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})}$ is also a CAT(κ)-space.[1]
4. Let (X_n,d_n) be a sequence of CAT(κ_n)-metric spaces where ${\displaystyle \lim _{n\to \infty }\kappa _{n}=-\infty .}$ Then the ultralimit ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n},p_{n})}$ is real tree.[1]

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 ${\displaystyle \mathbb {N} }$ and let p_n ∈ X be a sequence of base-points. Then the ω–ultralimit of the sequence ${\displaystyle (X,{\frac {d}{n}},p_{n})}$ is called the asymptotic cone of X with respect to ω and ${\displaystyle (p_{n})_{n}\,}$ and is denoted ${\displaystyle Cone_{\omega }(X,d,(p_{n})_{n})\,}$. 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 ${\displaystyle Cone_{\omega }(X,d)\,}$ or just ${\displaystyle Cone_{\omega }(X)\,}$.

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.[6] Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.[7]

1. Let (X,d) be a compact metric space and put (X_n,d_n)=(X,d) for every ${\displaystyle n\in \mathbb {N} }$. Then the ultralimit ${\displaystyle (X_{\infty },d_{\infty })=\lim _{\omega }(X_{n},d_{n})}$ 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 ${\displaystyle A_{1}=\{n|(X_{n},d_{n})=(X,d_{X})\}\,}$ and ${\displaystyle A_{2}=\{n|(X_{n},d_{n})=(Y,d_{Y})\}\,}$. Thus A₁, A₂ are disjoint and ${\displaystyle A_{1}\cup A_{2}=\mathbb {N} .}$ Therefore, one of A₁, A₂ has ω-measure 1 and the other has ω-measure 0. Hence ${\displaystyle \lim _{\omega }(X_{n},d_{n})}$ is isometric to (X,d_X) if ω(A₁)=1 and ${\displaystyle \lim _{\omega }(X_{n},d_{n})}$ is isometric to (Y,d_Y) if ω(A₂)=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
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) ${\displaystyle \lim _{\omega }(M,nd,p)}$ 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 ${\displaystyle \lim _{\omega }(M,nd,p)}$ is isometric to the Euclidean space ${\displaystyle \mathbb {R} ^{m}}$ with the standard Euclidean metric.[8]
4. Let ${\displaystyle (\mathbb {R} ^{m},d)}$ be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone ${\displaystyle Cone_{\omega }(\mathbb {R} ^{m},d)}$ is isometric to ${\displaystyle (\mathbb {R} ^{m},d)}$.
5. Let ${\displaystyle (\mathbb {Z} ^{2},d)}$ 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 ${\displaystyle Cone_{\omega }(\mathbb {Z} ^{2},d)}$ is isometric to ${\displaystyle (\mathbb {R} ^{2},d_{1})}$ where ${\displaystyle d_{1}\,}$ is the Taxicab metric (or L¹-metric) on ${\displaystyle \mathbb {R} ^{2}}$.
6. Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone ${\displaystyle Cone_{\omega }(X)\,}$ is a real tree.[1][9]
7. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone ${\displaystyle Cone_{\omega }(X)\,}$ is a single point.
8. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone ${\displaystyle Cone_{\omega }(X)\,}$ is also a CAT(0)-space.[1]

• 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.

• Ultrafilter
• Geometric group theory
• Gromov-Hausdorff convergence