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Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has finitely many ends. On the other hand, if we consider manifolds which have even compactly supported negative Ricci curvature, I believe it is not that difficult to produce examples with an infinite number of ends.

Now suppose we consider the case where $M$ has sectional curvature $K$ in between $-a^2$ and $-b^2$, where $a > b > 0$. In this case, is there a description of how many ends $M$ could have? Of course, I realize that nice enough simply connected spaces like $\mathbb{H}^n$ have only one end. I was wondering what is known outside this. I naively expect some statement like $M$ has lots of ends, unless $M$ is "really nice". Is this too naive?

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    $\begingroup$ The question is too broad. The complement of a Cantor set in the 2-sphere has a complete hyperbolic metric. In higher dimensions there are complete hyperbolic manifolds with finitely generated fundamental groups and infinitely many ends, see theorem 2b in [M. Kapovich, L. Potyagailo, On absence of Ahlfors' and Sullivan's finiteness theorems for Kleinian groups in higher dimensions], math.ucdavis.edu/~kapovich/EPR/KP_AFT_1991.pdf. On the other hand, if you assume that the complete negatively pinched manifold has finite volume, there is a complete classification of ends. $\endgroup$
    – Igor Belegradek
    Commented Apr 30, 2022 at 22:08
  • $\begingroup$ @IgorBelegradek Thanks. Are there nice sufficient conditions for hyperbolic manifolds with finitely generated fundamental groups to have infinitely many ends? Also, where can I find the classification of ends for finite volume pinched manifolds? $\endgroup$
    – Math_Learner
    Commented Apr 30, 2022 at 22:47
  • $\begingroup$ I don't think many people are interested in hyperbolic manifolds with infinitely many ends (except maybe in dimension 2), so not much is known beyond the above examples. Finite volume negatively pinched manifolds have finitely many ends each being the product of a closed infranilmanifold and a ray. This is Ballmann-Gromov-Schroeder's book "Manifolds of nonpositive curvature". Ontaneda showed that any finite collection of closed infranilmanifolds whose union bounds a smooth manifold can occur in the above. This gives a complete classification, see Theorem A in arxiv.org/abs/1406.1730. $\endgroup$
    – Igor Belegradek
    Commented Apr 30, 2022 at 23:38
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    $\begingroup$ @YCor Is it true that finitely generated groups of exponential growth have infinitely many ends? $\endgroup$
    – Math_Learner
    Commented May 1, 2022 at 8:33
  • $\begingroup$ @Math_Learner of course not. It is a theorem of Stallings that a torsion-free f.g. growth is infinitely-ended iff it splits as a nontrivial free product. (And every direct product of two infinite f.g. groups is 1-ended.) $\endgroup$
    – YCor
    Commented May 1, 2022 at 14:25

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