Connecting conformal dimension and Poincaré 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 Poincaré profiles $p_{\mathrm{\Lambda }}$ 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 ${\mathbb{R}}^n{⋊}_A\mathbb{R}$ where $A\in \text{GL}(n,\mathbb{R})$ is diagonalisable. A key tool is a lower bound for $p_{\mathrm{\Lambda }}$ 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 $G$, $p_{\mathrm{\Lambda }}(G)>1$ if and only if Benjamini–Schramm–Timár’s separation profile grows faster than $r^{\alpha }$ for some $\alpha >0$, if and only if Confdim$({\mathrm{\partial }}_{\mathrm{\infty }}G)>1$. 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.