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Suppose $G$ is a one-ended word-hyperbolic group and $\xi$ is a (local) cut point of $\partial G$. Fix any visual metric on $\partial G$ and let $U(\epsilon,\xi)$ be the connected component of $\xi$ inside $B(\epsilon,\xi)$, where $B(\epsilon,\xi)$ is the ball of radius $\epsilon$ around $\xi$ in $\partial G$. Let $N(\epsilon,\xi)$ be the number of connected components of $U(\epsilon,\xi)\setminus \xi$.

For a fixed $\xi$, if we let $\epsilon$ go to $0$, is $N(\epsilon,\xi)$ bounded?

(Presumably, it is the same to ask whether (Rips complexes on) horospheres in $\text{Cay}(G)$ must have finitely many ends.)

More generally, does $\limsup_\epsilon N(\epsilon,\xi)$ have some algebraic meaning in terms of $\mathbb{Z}$-splittings?

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    $\begingroup$ I think this follows from Bowditch's famous paper. Every local cut point is in a cut pair. If the cut pair is in a necklace, then $N(\epsilon,\xi)=2$ for small $\epsilon$. If not, the cut pair corresponds to a virtually cyclic edge group of the JSJ. In this case, for small $\epsilon$, $N$ is the sum of the indices of the incident edge groups. $\endgroup$
    – HJRW
    Commented Apr 14, 2017 at 5:02
  • $\begingroup$ (I'm not sure I have the terminology right, especially "necklace".) $\endgroup$
    – HJRW
    Commented Apr 14, 2017 at 5:03
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    $\begingroup$ Thanks Henry. It looks like Bowditch's paper answers this question on the nose. $\endgroup$
    – David Cohen
    Commented Apr 14, 2017 at 15:46
  • $\begingroup$ @HJRW could you make this an answer? $\endgroup$
    – YCor
    Commented Apr 21, 2017 at 21:30

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