Ultralimit

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.
An ultrafilter "ω" on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can 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 formula_1.
If formula_2 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 formula_3, if for every formula_4 we have:
It is not hard to see the following:
An important basic fact states that, if ("X","d") is compact and "ω" is a non-principal ultrafilter on formula_1, 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 formula_9 (as closed intervals are compact).
Ultralimit of metric spaces with specified base-points.
Let "ω" be a non-principal ultrafilter on formula_1. 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 formula_11, 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 formula_12.
Let us denote the set of all admissible sequences by formula_13.
It is easy to see from the triangle inequality that for any two admissible sequences formula_14 and formula_15 the sequence ("d""n"("xn","yn"))"n" is bounded and hence there exists an "ω"-limit formula_16. Let us define a relation formula_17 on the set formula_13 of all admissible sequences as follows. For formula_19 we have formula_20 whenever formula_21 It is easy to show that formula_17 is an equivalence relation on formula_23
The ultralimit with respect to "ω" of the sequence ("X""n","d""n", "p""n") is a metric space formula_24 defined as follows.
As a set, we have formula_25 .
For two formula_17-equivalence classes formula_27 of admissible sequences formula_14 and formula_15 we have formula_30
It is not hard to see that formula_31 is well-defined and that it is a metric on the set formula_32.
Denote formula_33 .
On basepoints in the case of uniformly bounded spaces.
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("X""n")≤"C" for every formula_34. Then for any choice "pn" of base-points in "Xn" "every" sequence formula_35 is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit formula_24 depends only on ("Xn","dn") and on "ω" but does not depend on the choice of a base-point sequence formula_37. In this case one writes formula_38.
Basic properties of ultralimits.
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.
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 formula_1 and let "pn" ∈ "X" be a sequence of base-points. Then the "ω"–ultralimit of the sequence formula_48 is called the asymptotic cone of "X" with respect to "ω" and formula_49 and is denoted formula_50. 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 formula_51 or just formula_52.
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. Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.