In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces
The concept captures the limiting behavior of finite configurations in the
spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence.
Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.
An ultrafilter, denoted as 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 also which, given any subset X of
is non-principal if it contains no finite set.
In the following, ω is a non-principal ultrafilter on
is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called ω-limit of xn, denoted as
it holds that It is observed that, A fundamental fact[1] states that, if (X,d) is compact and ω is a non-principal Ultrafilter on
, 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
, as closed intervals are compact.
Let (Xn ,dn) be a sequence of metric spaces with specified base-points pn ∈ Xn.
If the sequence of real numbers (dn(xn ,pn))n is bounded, that is, if there exists a positive real number C such that
, then denote the set of all admissible sequences by
It follows from the triangle inequality that for any two admissible sequences
the sequence (dn(xn,yn))n is bounded and hence there exists an ω-limit
The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space
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
Then for any choice pn of base-points in Xn every sequence
depends only on (Xn,dn) and on ω but does not depend on the choice of a base-point sequence
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] 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
is called the asymptotic cone of X with respect to ω and
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
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.