Connecting conformal dimension and Poincar\'e profiles

We strengthen the connection between the Ahlfors-regular (AR) conformal dimension Confdim$(Z)$ of a compact AR metric space $Z$ and a certain critical exponent of the Poincar\'e profiles $p_{\Lambda}$ of its hyperbolic cone $X$ in the sense of Bonk--Schramm. We prove that the two values are equal in two situations: firstly, when $Z$ is a product $C\times [0,1]$ where $C$ is a compact AR metric space; and secondly when $X$ is quasi-isometric to a Heintze manifold $\mathbb R^n\rtimes_A\mathbb R$ where $A\in\textrm{GL}(n,\mathbb R)$ is diagonalisable. A key tool is a lower bound for $p_{\Lambda}$ for combinatorial round trees which also applies to various random group models and families of Coxeter groups. We also show that for a torsion free hyperbolic group $G$, $p_{\Lambda}(G)>1$ if and only if Benjamini--Schramm--Tim\'ar's separation profile grows faster than $r^\alpha$ for some $\alpha>0$, if and only if Confdim$(\partial_\infty G)>1$. On the other hand, we find new, non-virtually-Fuchsian examples of groups with the same separation profile as $\mathbb{H}^2$. All these results imply various obstructions to coarse and regular embeddings of such groups.