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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ultralimit is a geometric construction that assigns to a sequence of
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s ''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 In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion of
Gromov–Hausdorff convergence In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. Gromov–Hausdorff distance The Gromov–Hausdorff ...
of metric spaces.


Ultrafilters

An
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
''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 In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
(''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 space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s 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''(''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)_ and \mathbf y=(y_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 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
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 In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s 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 ''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

#If (''Xn'',''dn'') are
geodesic metric space In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...
s then (X_\infty, d_\infty)=\lim_\omega(X_n, d_n, p_n) is also a geodesic metric space. #If (''Xn'',''dn'') are
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
s 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 ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to ...
, Vol. 89(1984), pp. 349–374.
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. #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'') are
proper metric space This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provid ...
s and that p_n\in 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 3-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_\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 ''pn'' ∈ ''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, ''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 Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups ...
since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. T ...
invariants of metric spaces in general and of finitely generated groups in particular.John Roe. ''Lectures on Coarse Geometry.''
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
, 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'',''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=\\, 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'',''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 connected
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...
of dimension ''m'', where ''g'' is a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
on ''M''. Let ''d'' be the metric on ''M'' corresponding to ''g'', so that (''M'',''d'') is a
geodesic metric space In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...
. 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 In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
''TpM'' of ''M'' at ''p'' with the distance function on ''TpM'' given by the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often de ...
''g(p)''. Therefore, the ultralimit \lim_\omega(M,n d, p) is isometric to the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
\mathbb R^m with the standard
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
. #Let (\mathbb R^m, d) be the standard ''m''-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
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 The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
, 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 The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
, 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 ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to ...
, 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 ''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherch ...
. Volume 86, Number 1, December 1997, pp. 115–197.


See also

*
Ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
*
Geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups ...
* Gromov-Hausdorff convergence Geometric group theory Metric geometry