Ultralimit

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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 Xn a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces Xn 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[edit]

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}\to \{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 F\subseteq \mathbb N we have ω(F)=0.

Limit of a sequence of points with respect to an ultrafilter[edit]

Let ω be 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 xX, the point x is called the ω -limit of xn, 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\to\infty} 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[1] 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 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[edit]

Let ω be a non-principal ultrafilter on \mathbb N . Let (Xn,dn) be a sequence of metric spaces with specified base-points pnXn.

Let us say that a sequence (x_n)_{n\in\mathbb N}, where xnXn, is admissible, if the sequence of real numbers (dn(xn,pn))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)_{n\in\mathbb N} and \mathbf y=(y_n)_{n\in\mathbb N} the sequence (dn(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 of all admissible sequences as follows. For \mathbf x, \mathbf y\in \mathcal A we have \mathbf x\sim\mathbf 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 (Xn,dn, pn) is a metric space (X_\infty, d_\infty) defined as follows.[2]

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)_{n\in\mathbb N} and \mathbf y=(y_n)_{n\in\mathbb 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[edit]

Suppose that (Xn,dn) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C>0 such that diam(Xn)≤C for every n\in \mathbb N. Then for any choice pn of base-points in Xn 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 (Xn,dn) 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[edit]

  1. If (Xn,dn) are geodesic metric spaces then (X_\infty, d_\infty)=\lim_\omega(X_n, d_n, p_n) is also a geodesic metric space.[1]
  2. If (Xn,dn) are complete metric spaces then (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 (Xn,dn) is a repeating sequence of a space (X,d) which is not complete.[5]

  1. 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).
  2. Suppose that (Xn,dn) are proper metric spaces and that p_n\in X_n are base-points such that the pointed sequence (Xn,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).[1]
  3. 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.[1]
  4. Let (Xn,dn) be a sequence of CAT(κn)-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.[1]

Asymptotic cones[edit]

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 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) 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[edit]

  1. Let (X,d) be a compact metric space and put (Xn,dn)=(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).
  2. 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)=(Y,d_Y)\}\,. Thus A1, A2 are disjoint and A_1\cup A_2=\mathbb N. Therefore one of A1, A2 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(A1)=1 and \lim_\omega(X_n,d_n) is isometric to (Y,dY) 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 pM. Then the ultralimit (and even the ordinary Gromov-Hausdorff limit) \lim_\omega(M,n d, p) is isometric to the tangent space TpM of M at p with the distance function on TpM 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.[8]
  4. 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).
  5. 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.
  6. 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.[1][9]
  7. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone Cone_\omega(X)\, is a single point.
  8. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone Cone_\omega(X)\, is also a CAT(0)-space.[1]

Footnotes[edit]

  1. ^ a b c d e f g 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
  2. ^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; Definition 7.19, p. 107.
  3. ^ 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.
  4. ^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; Proposition 7.20, p. 108.
  5. ^ Bridson, Haefliger "Metric Spaces of Non-positive curvature" Lemma 5.53
  6. ^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2
  7. ^ 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.
  8. ^ 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
  9. ^ John Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003. ISBN 978-0-8218-3332-2; Example 7.30, p. 118.

Basic References[edit]

  • 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[edit]