Connecting conformal dimension and Poincar\'e profiles

TL;DR

This paper explores the relationship between conformal dimension and Poincaré profiles in hyperbolic spaces, establishing equalities in specific cases and deriving implications for group embeddings and geometric properties.

Contribution

It proves the equality of conformal dimension and Poincaré profile exponents in new settings and introduces bounds for combinatorial structures, advancing understanding of hyperbolic group boundaries.

Findings

Conformal dimension equals Poincaré profile exponent in certain product and quasi-isometric cases.

Lower bounds for Poincaré profiles are established for combinatorial round trees.

Characterization of hyperbolic groups with specific separation profile growth and conformal dimension.

Abstract

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.