  In this paper, we study the existence of traces for Sobolev spaces on the
hyperbolic filling $X$ of a compact metric space $Z$ equipped with a doubling
measure. Given a suitable metric on $X$, we can regard $Z$ as the boundary of
$X$. After equipping $X$ with a weighted measure $\mu$ via the measure on $Z$
and the Euclidean arc length, we give characterizations for the existence of
traces for first-order Sobolev spaces.
