# Inverse limit in metric geometry

Question. Did you ever see inverse limits to be used (or even seriousely considered) anywhere in metric geometry (but NOT in topology)?

The definition of inverse limit for metric spaces is given below. (It is usual inverse limit in the category with class of objects formed by metric spaces and class of morphisms formed by short maps.)

Definition. Consider an inverse system of metric spaces $X_n$ and short maps $\phi_{m,n}:X_m\to X_n$ for $m\ge n$; i.e.,(1) $\phi_{m,n}\circ \phi_{k,m}=\phi_{k,n}$ for any triple $k\ge m\ge n$ and (2) for any $n$, the map $\phi_{n,n}$ is identity map of $X_n$.

A metric space $X$ is called inverse limit of the system $(\phi_{m,n}, X_n)$ if its underlying space consists of all sequences $x_n\in X_n$ such that $\phi_{m,n}(x_m)=x_n$ for all $m\ge n$ and for any two such sequences $(x_n)$ and $(y_n)$ the distance is defined by

$$| (x_n) (y_n)| = \lim_{n\to\infty} | x_n y_n | .$$

Why: I have a theorem, with little cheating you can stated it this way: The class of metric spaces which admit path-isometries to Euclidean $d$-spaces coincides with class of inverse limits of $d$-polyhedral spaces. In the paper I write: it seems to be the first case when inverse limits help to solve a natural problem in metric geometry. But I can not be 100% sure, and if I'm wrong I still have time to change this sentence.

• Short map = nonexpansive = Lipschitz constant at most one? – Bill Johnson Feb 21 '10 at 1:40
• Bill: yes . – Tom Leinster Feb 21 '10 at 1:58
• Anton: as I understand it, "inverse system" means either functor from a cofiltered category, or functor from a (co)directed poset, according to taste (and then "inverse limit" means limit). Just to be clear, are you particularly interested in this special case of limits? Or are any types of limit of metric space interesting for you? – Tom Leinster Feb 21 '10 at 2:04
• I think the topology induced by the metric is not the inverse limit topology. But maybe this is no problem. It just irratated me. – HenrikRüping Feb 21 '10 at 13:20
• There is another irritating example: Consider the inverse system $X_n=\\{-n,\ldots,n\\}$ with distance $d(m,m')=|m-m'|$ and projections $\varphi_{n+1,n}:\\{-(n+1),\ldots,n+1\\}\rightarrow \\{-n,\ldots,n\\}$ defined by $\varphi_{n+1,n}(-n-1)=-n,\quad \varphi_{n+1,n}(n+1)=n,\quad \varphi_{n+1,n}|_{\\{-n,\ldots n\\}}=id$. Then the inverse limit is $\mathbb{Z}\cup\\{\pm \infty\\}$ and the metric is really just a quasimetric. – HenrikRüping Feb 21 '10 at 13:47

This paper by P.-E. Caprace, uses a "refined boundary" of a CAT(0) space. This boundary is constructed in the following way : given a point $\xi$ in the boundary at infinity of your space $X$, you construct a point $X_\xi$, which is the inverse limit of the horoballs centered at $\xi$. Here the maps $\phi_{m,n}$ are the CAT(0) projections. Then the space $X_\xi$ is itself CAT(0), and you can iterate the construction. Under reasonable hypotheses, the construction stops after a finite number of steps, and the refined boundary is the union of all the spaces you get.

(In the case of symmetric spaces, this construction has been already considered by Karpelevic in 1965, but with different definitions, and I don't think he saw it as inverse limits).

Take a look at this paper of Irwin and Solecki which, I think, is relevant to your question. They use a model theoretic language for inverse limits (projective Fraisse limits) and use this machinery to derive a proof of the surjective universality of the pseudo-arc among chainable continua.

The paper has appeared in: Trans. Amer. Math. Soc. 358 (2006), 3077-3096.

• Accidentally, that is the same paper suggested by François G. Dorais. This is topology and this is NOT metric geometry... – Anton Petrunin Feb 21 '10 at 14:47
• Oh, thanks, I didn't look carefully at the comments. Do you want me to delete the answer? – Pandelis Dodos Feb 21 '10 at 14:50
• @Pandelis. No, why should I? – Anton Petrunin Feb 21 '10 at 15:11
• You are right, why should you? :-) By the way I have a similar question in Banach space theory. While inductive limits have been extensively used I am not aware whether an inverse limit costruction has been considered to build quotients of Banach spaces with special properties. – Pandelis Dodos Feb 21 '10 at 15:26

I am not sure, if you are still looking for an example, and whether you feel that my examples are artificial.

Every locally compact, almost connected group is a projective limit of Lie groups, in particular a projective limit of metrizable groups. The limit becomes metrizable itself, if and only if the limit is countable. There is a book by Hofmann and Morris about pro-Liegroups, which studies projective limits of Lie groups.

Also the vector space $C_b^\infty(X) = \cap_k C_b^k(X)$ of smooth bounded functions is a projective limit of normed spaces.

I'd be surprised if there were any applications, for the simple reason that your definition of inverse limit for metric geometry doesn't translate well into topology. It's easy to disconnect points in the limit which aren't disconnected in the terms.

For example, fix a metric space (X, d) and define (Xn, dn) = (X, 2nd), with φm, n the identity on the underlying set X. Then the inverse limit would have underlying set X (essentially), but d(x, y) = ∞ unless x = y.

Come to think of it, this seems to provide a counterexample to your statement, since the underlying space could easily be a polyhedron (with uncountable underlying set), while any discrete set which embeds into Rd is at most countable.

(Never mind, I see in your paper that you assume the space to be embedded is compact.)