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Connecting conformal dimension and Poincaré profiles

David Hume, John M. Mackay

TL;DR

This work sharpens the link between the AR conformal dimension $\mathrm{Confdim}(Z)$ of a compact AR space $Z$ and the critical exponent $p_{\Lambda}$ of Poincaré profiles for the hyperbolic cone $\mathrm{Con}(Z)$, proving equality in the product case $Z\times[0,1]$ and for hyperbolic cones quasi-isometric to diagonal Heintze groups. The authors introduce combinatorial round trees and derive precise lower bounds for $p_{\Lambda}$, enabling new non-embedding obstructions and expanding applications to random groups and Coxeter groups. They also analyze conformal-dimension-one phenomena in hyperbolic groups: either separation profiles grow like a positive power, or the group is virtually Fuchsian, with new non-virtually-Fuchsian examples sharing the same separation as $\mathbb{H}^2$. Collectively, these results provide a versatile framework linking boundary conformal geometry to large-scale coarse geometric properties, and yield concrete obstructions to coarse and regular embeddings across a range of group families.

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 Poincaré profiles $p_Λ$ 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 $p_Λ$ 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 $G$, $p_Λ(G)>1$ if and only if Benjamini--Schramm--Timár's separation profile grows faster than $r^α$ for some $α>0$, 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.

Connecting conformal dimension and Poincaré profiles

TL;DR

This work sharpens the link between the AR conformal dimension of a compact AR space and the critical exponent of Poincaré profiles for the hyperbolic cone , proving equality in the product case and for hyperbolic cones quasi-isometric to diagonal Heintze groups. The authors introduce combinatorial round trees and derive precise lower bounds for , enabling new non-embedding obstructions and expanding applications to random groups and Coxeter groups. They also analyze conformal-dimension-one phenomena in hyperbolic groups: either separation profiles grow like a positive power, or the group is virtually Fuchsian, with new non-virtually-Fuchsian examples sharing the same separation as . Collectively, these results provide a versatile framework linking boundary conformal geometry to large-scale coarse geometric properties, and yield concrete obstructions to coarse and regular embeddings across a range of group families.

Abstract

We strengthen the connection between the Ahlfors-regular (AR) conformal dimension Confdim of a compact AR metric space and a certain critical exponent of the Poincaré profiles of its hyperbolic cone in the sense of Bonk--Schramm. We prove that the two values are equal in two situations: firstly, when is a product where is a compact AR metric space; and secondly when is quasi-isometric to a Heintze manifold where is diagonalisable. A key tool is a lower bound for 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 , if and only if Benjamini--Schramm--Timár's separation profile grows faster than for some , if and only if Confdim. On the other hand, we find new, non-virtually-Fuchsian examples of groups with the same separation profile as . All these results imply various obstructions to coarse and regular embeddings of such groups.

Paper Structure

This paper contains 16 sections, 34 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main concepts
  3. Results on critical exponents
  4. Results at conformal dimension $1$
  5. Outline
  6. Notation
  7. Critical exponents
  8. Round trees
  9. Poincaré constants of round trees
  10. Applications of regular round trees
  11. Product spaces
  12. Heintze manifolds $\mathbb{R}^n\rtimes_{a_1,\ldots,a_n}\mathbb{R}$
  13. Random groups
  14. (Relatively) hyperbolic Coxeter groups
  15. Poincaré profiles detect hyperbolic groups of conformal dimension $1$
  16. ...and 1 more sections

Key Result

Theorem 1.6

Let $Z$ be a compact Ahlfors regular metric space and let $\mathop{\mathrm{Con}}\nolimits(Z)$ be the hyperbolic cone of $Z$. Then

Figures (4)

  • Figure 1: A space $X$ with $\Lambda_{\pi_1(X)}^1(r)\simeq\mathop{\mathrm{sep}}\nolimits_{\pi_1(X)}(r) \simeq \log(r)$.
  • Figure 2: Part of a round tree graph
  • Figure 3: Constructing $C$ (the bold lines) in two of the components of $U\setminus\gamma'$
  • Figure 4: Three types of component of $U\setminus C$, with $Y'\setminus Y"$ highlighted in pale grey and $Y"$ in a darker grey

Theorems & Definitions (62)

  • Definition 1.1: HumeMackTess-Pprof
  • Definition 1.2
  • Theorem 1.6: HumeMackTess-PprofLie
  • Theorem 1.7
  • proof
  • Theorem 1.8: Theorem \ref{['thm:product-crit-exponent']}
  • Corollary 1.9: Corollary \ref{['cor:crit-exponent-heintze']}
  • Theorem 1.11
  • Remark 1.12
  • Corollary 1.13
  • ...and 52 more