Ultralimit

In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces ${\displaystyle X_{n}}$. The concept captures the limiting behavior of finite configurations in the ${\displaystyle X_{n}}$ 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 ${\displaystyle \mathbb {N} }$ is a set of nonempty subsets of ${\displaystyle \mathbb {N} }$ (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 ${\displaystyle \mathbb {N} }$, contains either X or ${\displaystyle \mathbb {N} }$\ X. An ultrafilter on ${\displaystyle \mathbb {N} }$ 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 ${\displaystyle \mathbb {N} }$.

If ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called ω-limit of x_n, denoted as ${\displaystyle x=\lim _{\omega }x_{n}}$, if for every ${\displaystyle \epsilon >0}$ it holds that

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

It is observed that,

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

A fundamental 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.

Ultralimit of metric spaces with specified base-points

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.

Suppose 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 ${\displaystyle d_{n}(x_{n},p_{n})\leq C}$, then denote the set of all admissible sequences by ${\displaystyle {\mathcal {A}}}$.

It follows 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 ${\displaystyle {\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} ):=\lim _{\omega }d_{n}(x_{n},y_{n})}$. One can define a relation ${\displaystyle \sim }$ on the set ${\displaystyle {\mathcal {A}}}$ of all admissible sequences as follows. For ${\displaystyle \mathbf {x} ,\mathbf {y} \in {\mathcal {A}}}$, there is ${\displaystyle \mathbf {x} \sim \mathbf {y} }$ whenever ${\displaystyle {\hat {d}}_{\infty }(\mathbf {x} ,\mathbf {y} )=0.}$ This helps 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]

Written as a set, ${\displaystyle X_{\infty }={\mathcal {A}}/{\sim }}$ .

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

This shows that ${\displaystyle d_{\infty }}$ is well-defined and that it is a metric on the set ${\displaystyle X_{\infty }}$.

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

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

Basic properties of ultralimits

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]

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 ${\displaystyle \mathbb {N} }$ 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]

Examples

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 base point 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]

Footnotes

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

• Geometric group theory