The authors show that if every peripheral subgroup of a relatively hyperbolic group embeds into the product of finitely many trees, then the entire group does. Moreover they give explicit bounds for the number of trees required. The analogue results were proved for hyperbolic groups by S. Buyalo and N. Lebedeva [St. Petersbg. Math. J. 19, No. 1, 45-65 (2008); translation from Algebra Anal. 18, No. 1, 60-92 (2007; Zbl 1145.54029)] and S. Buyalo, A. Dranishnikov and V. Schroeder [Invent. Math. 169, No. 1, 153-192 (2007; Zbl 1157.57003)].
The main result of the paper under review is the following theorem: Suppose the group \(G\) is hyperbolic relative to subgroups \(H_1,H_2,\ldots,H_n\). If each \(H_i\) quasi-isometrically embeds into a product of \(m\) metric trees, then \(\text{asdim}(G)<\infty\) and \(G\) quasi-isometrically embeds into a product of \(M\) metric trees, where \(M=\max\{\text{asdim}(G),m+1\}+m+1<\infty\). Conversely, if \(G\) quasi-isometrically embeds into a product of \(N\) metric trees, then each peripheral group does also.
As an application the authors prove the following theorem: Let \(G=\pi_1(M)\), where \(M\) is a compact, orientable 3-manifold whose (possible empty) boundary is a union of tori. Then \(\text{eco-dim}(G)<\infty\) if and only if no manifold in the prime decomposition of \(M\) has NIL geometry; in this case, \(\text{eco-dim}(G)\leq 8\). – Here \(\text{eco-dim}(X)\), for a metric space \(X\), denotes the smallest \(n\in\mathbb{N}\) so that \(X\) quasi-isometrically embeds in the product of \(n\) metric trees, and set \(\text{eco-dim}(X)=\infty\) if no such embedding exists.