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I'm looking for a definition of “ends” of a metric space that is well-defined even for non geodesic or locally finite metric spaces, invariant under quasi-isometries (or more generally coarse isometries), and that subsumes the usual definition for graphs. A good “test” for such a definition would be whether it satisfies the Hopf-Freudenthal result that the number of ends of a transitive graph is either of $0,1,2,\infty$. I would like such a definition to be as “native” to coarse geometry as possible: reducing to graphs feels a bit cheaty.

A typical procedure would be to look at the ends of increasing Rips complexes on the space, but I'm not sure this is well-defined (or rather, behaves well), since the Rips complexes wouldn't necessarily yield connected graphs, and the path metric on the Rips complex wouldn't have much similarity with the original metric on the space.

So, let me propose the following, until a better definition comes along:

Let $(X,d)$ be a metric space. A subset $A$ of $X$ is $r$-connected ($r>0$) if any two points $x,y \in A$ of this subset can be connected by a sequence $x=z_0,\dots,z_n=y$ with $z_i \in A$ and $d(z_i,z_{i+1}) \leq r$.

The $r$-ends of $X$ are constructed as the limit of the $r$-connected components of infinite diameter of $X-K$, for $K$ ranging over all subsets of $X$ of finite diameter. If I'm not mistaken, there is a way to map the $r$-ends of $X$ to its $s$-ends, for $r<s$, and one could define the “coarse ends” of $X$ as the appropriate limit.

Note that I think we shouldn't let $K$ up there range over all subsets of $X$ that are of finite diameter and $r$-connected, since this doesn't give us a directed system anymore.

Note also that the $r$-ends themselves don't necessarily satisfy Hopf-Freudental, again because of this possible non-$r$-connectedness.

The main thing I'm queasy about in this construction is the fact that it mixes two metrics essentially: the original one on $X$ and the one defined by $r$-connectedness… this looks fishy.

That's it: I believe there should be a “clearly correct” definition, but I'm not sure which it is. Thanks!

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    $\begingroup$ A general definition for a coarse space $X$ is to consider the Stone space of the Boolean algebra of functions $f:X\to\{0,1\}$ such that $\{(x,y):f(x)\neq f(y)\}$ is an entourage. For instance for a connected locally finite graph, with the metric coarse structure, this is the "usual" space of ends. For an arbitrary group this also recovers the "usual" notion (the familiar Freudenthal one for f.g. groups and the 1950 Specker one for general groups). Note that even for a countable group this is not always a metrizable space. (This is also called binary Higson corona.) $\endgroup$
    – YCor
    Commented Mar 9, 2022 at 16:49
  • $\begingroup$ This seems like a very good candidate, thanks! Is there any all-in-one reference for this binary Higson corona? I assume Roe's book is a good start. Edit: Ah, I just found this I. V. Protasov, Binary coronas of balleans, Algebra Discrete Math., 2003, Issue 4, 50–65. $\endgroup$
    – user148575
    Commented Mar 10, 2022 at 8:16

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