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) $omega:2^\{mathbb\; N\}\; o\; \{0,1\}$ 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 $Fsubseteq\; mathbb\; N$ 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 $(x\_n)\_\{nin\; mathbb\; N\}$ 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::$omega\{n:\; d(x\_n,x)le\; epsilon\; \}=1.$

It is not hard to see the following:
* If an "ω" -limit of a sequence of points exists, it is unique.
* If $x=lim\_\{n\; oinfty\}\; x\_n$ 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 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 $(x\_n)\_\{ninmathbb\; N\}$, where "x"_"n"∈"X"_"n", is "admissible", if the sequence of real numbers ("d"_"n"("x_n","y_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 $mathbf\; x=(x\_n)\_\{ninmathbb\; N\}$ and $mathbf\; y=(y\_n)\_\{ninmathbb\; N\}$ 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 $mathbf\; x,\; mathbf\; yin\; mathcal\; A$ we have $mathbf\; xsimmathbf\; y$ 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. [John Roe. "Lectures on Coarse Geometry." American Mathematical Society, 2003. ISBN-13: 9780821833322; Definition 7.19, p. 107.]

As a set, we have $X\_infty=mathcal\; A/sim$ .

For two $sim$-equivalence classes $[mathbf\; x]\; ,\; [mathbf\; y]$ of admissible sequences $mathbf\; x=(x\_n)\_\{ninmathbb\; N\}$ and $mathbf\; y=(y\_n)\_\{ninmathbb\; N\}$ 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 $nin\; mathbb\; N$. Then for any choice "p_n" of base-points in "X_n" "every" sequence $(x\_n)\_n,\; x\_nin\; X\_n$ 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 $p\_nin\; X\_n.$. 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.
#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.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-13: 9780821833322; Proposition 7.20, p. 108.]
#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 $p\_nin\; X\_n$ 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. 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]
#Let ("X_n","d_n") be a sequence of CAT("κ_n")-metric spaces where $lim\_\{n\; oinfty\}kappa\_n=-infty.$ 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.John Roe. "Lectures on Coarse Geometry." American Mathematical Society, 2003. ISBN-13: 9780821833322] Asymptotic cones turned also out to be a useful tool in the study of relatively hyperbolic groups and their generalizations. [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.]

Examples

#Let ("X","d") be a compact metric space and put ("X"_"n","d"_"n")=("X","d") for every $nin\; mathbb\; N$. 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 "A"₁, "A"₂ are disjoint and $A\_1cup\; A\_2=mathbb\; N.$ 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 basepoint "p"∈"M". Then the ultralimit (and even the ordinary Gromov-Hausdorff limit) $lim\_omega(M,n\; d,\; p)$ 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 $lim\_omega(M,n\; d,\; p)$ is isometric to the Euclidean space $mathbb\; R^m$ with the standard Euclidean metric. [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 ]
#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. [John Roe. "Lectures on Coarse Geometry." American Mathematical Society, 2003. ISBN-13: 9780821833322; Example 7.30, p. 118.]
#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

Basic References

*John Roe. "Lectures on Coarse Geometry." American Mathematical Society, 2003. ISBN-13: 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-13: 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.

ee also

*Ultrafilter
*Geometric group theory
*Gromov-Hausdorff convergence