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It is proved that a parameterized curve in a metric space $X$ is absolutely continuous if and only if its composition with any Lipschitz function on $X$ is absolutely continuous., Comment: 7 pages

We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmogorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different behaviours depending on the rank of the non-compact factor. For rank at least 2, we construct thick embeddings of $N$-vertex graphs with volume $CN\ln(1+N)$ and prove that this is optimal. For rank at most $1$ we prove lower bounds of the form $cN^a$ for some (explicit) $a>1$ which depends on the dimension of the Euclidean factor and the conformal dimension of the boundary of the non-compact factor. The main tool is a coarse geometric analogue of a thick embedding called a coarse wiring, with the key property that the minimal volume of a thick embedding is comparable to the ``minimal volume'' of a coarse wiring for symmetric spaces of dimension at least $3$. In the appendix it is proved that for each $k\geq 3$ every bounded degree graph admits a coarse wiring into $\mathbb{R}^k$ with volume at most $CN^{1+\frac{1}{k-1}}$. As a corollary, the same upper bound holds for real hyperbolic space of dimension $k+1$ and in both cases this result is optimal., Comment: 30 pages. Appendix by Larry Guth and Elia Portnoy. Accepted for publication in Transactions of the AMS