- Ultralimit
:"For the direct limit of a sequence of ultrapowers, see
Ultraproduct ."Inmathematics , an ultralimit is a geometric construction that assigns to a sequence ofmetric space s "Xn" a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces "Xn" and uses anultrafilter to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion ofGromov-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 thepower 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 space s 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"("xn","yn"))"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"("xn","yn"))"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 ("Xn","dn") is a sequence of
metric space s 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 "pn" of base-points in "Xn" "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 ("Xn","dn") 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 ("Xn","dn") are
geodesic metric space s then X_infty, d_infty)=lim_omega(X_n, d_n, p_n) is a also geodesic metric space.
#If ("Xn","dn") arecomplete metric space s 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 ("Xn","dn") are compact metric spaces that converge to a compact metric space ("X","d") in the Gromov-Hausdorff sense (this automatically implies that the spaces ("Xn","dn") have uniformly bounded diameter), then the ultralimit X_infty, d_infty)=lim_omega(X_n,d_n) is isometric to ("X","d").
#Suppose that ("Xn","dn") areproper metric space s and that p_nin X_n are base-points such that the pointed sequence ("X""n","dn","pn") 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 ("Xn","dn") 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 ("Xn","dn") be a sequence of CAT("κn")-metric spaces where lim_{n oinfty}kappa_n=-infty. Then the ultralimit X_infty, d_infty)=lim_omega(X_n,d_n, p_n) isreal tree .Asymptotic cones
An important class of ultralimits are the so-called "
asymptotic cone s" of metric spaces. Let ("X","d") be a metric space, let "ω" be a non-principal ultrafilter on mathbb N and let "pn" ∈ "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 "pn"="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) providequasi-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 ofrelatively hyperbolic group s 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","dX") and ("Y","dY") be two distinct compact metric spaces and let ("Xn","dn") be a sequence of metric spaces such that for each "n" either ("Xn","dn")=("X","dX") or ("Xn","dn")=("Y","dY"). 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","dX") if "ω"("A"1)=1 and lim_omega(X_n,d_n) is isometric to ("Y","dY") 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 connectedRiemannian manifold of dimension "m", where "g" is aRiemannian metric on "M". Let "d" be the metric on "M" corresponding to "g", so that ("M","d") is ageodesic metric space . Choose a basepoint "p"∈"M". Then the ultralimit (and even the ordinaryGromov-Hausdorff limit ) lim_omega(M,n d, p) is isometric to thetangent space "TpM" of "M" at "p" with the distance function on "TpM" given by theinner product "g(p)". Therefore the ultralimit lim_omega(M,n d, p) is isometric to theEuclidean space mathbb R^m with the standardEuclidean 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"-dimensionalEuclidean 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-dimensionalinteger 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 theTaxicab 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 areal 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
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