asymptotic cone

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

An ultrafilter ''ω'' on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and which, given any subset ''X'' of , contains either ''X'' or . An ultrafilter ''ω'' on is ''non-principal'' if it contains no finite set.

Limit of a sequence of points with respect to an ultrafilter

Let ''ω'' be a non-principal ultrafilter on $\mathbb N$.
If $(x_n)_$ 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:
:$\\in\omega.$

It is not hard to see the following:
* If an ''ω'' -limit of a sequence of points exists, it is unique.
* If $x=\lim_ x_n$ 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 compact and ''ω'' is a non-principal ultrafilter on $\mathbb N$, 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 $\mathbb R$ (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 $(x_n)_$, 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 $d_n(x_n,p_n)\le C$.
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 $\mathbf x=(x_n)_$ and $\mathbf y=(y_n)_$ 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 $\mathcal A$ 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 (''X''_''n'',''d''_''n'', ''p''_''n'') is a metric space $(X_\infty, d_\infty)$ defined as follows.

As a set, we have $X_\infty=\mathcal A/$ .

For two $\sim$-equivalence classes mathbf x mathbf y/math> of admissible sequences $\mathbf x=(x_n)_$ and $\mathbf y=(y_n)_$ 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 $n\in \mathbb N$. Then for any choice ''p_n'' of base-points in ''X_n'' ''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 (''X_n'',''d_n'') 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

#If (''X_n'',''d_n'') are geodesic metric spaces then $(X_\infty, d_\infty)=\lim_\omega(X_n, d_n, p_n)$ is also a 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 also a 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.
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.
#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 $p_n\in X_n$ 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 $(X_\infty, d_\infty)=\lim_\omega(X_n,d_n,p_n)$ 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 3-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 $\lim_\kappa_n=-\infty.$ 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 $(X, \frac, p_n)$ 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. Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.

Examples

#Let (''X'',''d'') be a compact metric space and put (''X''_''n'',''d''_''n'')=(''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'').
#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 $A_1=\\,$ and $A_2=\\,$. Thus ''A''₁, ''A''₂ are disjoint and $A_1\cup A_2=\mathbb N.$ 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.
#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. ; 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

References

*John Roe. ''Lectures on Coarse Geometry.'' American Mathematical Society, 2003. ; 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. ; 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

*Ultrafilter
* Geometric group theory
* Gromov-Hausdorff convergence

Geometric group theory
Metric geometry