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Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial X\to\mathbb{C}$. Is it possible to extend $f$ to a complex-valued continuous function on $X\cup\partial X$?

In particular, I am interested in the case where $X$ is the Cayley graph of a discrete hyperbolic group.

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    $\begingroup$ I don't know anything about the setting, but does the Tietze extension theorem apply here? en.wikipedia.org/wiki/Tietze_extension_theorem $\endgroup$
    – Thomas Rot
    Commented Mar 16, 2016 at 12:38
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    $\begingroup$ A possible strategy would be to associate to each point $x\in X$ a probability measure $\mu_x$ on $\partial X$, such that the measure depends continuously on the point and converges to $\delta_\zeta$ when $x$ converges to $\zeta\in\partial X$. I think such construction exist at least in particular cases (e.g. limit of the uniform measure on spheres, or using the critical exponent maybe). Then you extend $f$ by $\int f d\mu_x$. $\endgroup$
    – Benoît Kloeckner
    Commented Mar 16, 2016 at 12:57
  • $\begingroup$ @ThomasRot: Thanks for the advice, I didn't think about it. I'll work on it. $\endgroup$
    – EM90
    Commented Mar 17, 2016 at 9:54
  • $\begingroup$ @BenoîtKloeckner: Are you referring to something similar to the Patterson-Sullivan construction? (I'm not really into it, so maybe I'd better deep my knowledge about it. Thanks for reading, thanks for the advice.) $\endgroup$
    – EM90
    Commented Mar 17, 2016 at 9:54
  • $\begingroup$ Yes, I think that is what I had in mind (I have not worked with this myself, I just remembered that Besson-Courtois-Gallot have used these measure to prove their celebrated rigidity result). The reference Pacific J. Math. Volume 159, Number 2 (1993), 241-270 by Coornaert seems relevant by I have trouble retrieving it to check. $\endgroup$
    – Benoît Kloeckner
    Commented Mar 17, 2016 at 13:29

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