Ahlfors Regular Conformal Dimension

• Ahlfors regular conformal dimension is defined as the infimum of Hausdorff dimensions over all Ahlfors regular metrics quasisymmetric to a given space, ensuring controlled volumetric regularity.
• It is computed via combinatorial modulus methods where the critical exponent marks the threshold at which the p-modulus vanishes, linking discrete structures to continuum geometry.
• Attainment of this dimension indicates rigidity phenomena in settings such as hyperbolic groups and dynamical systems, distinguishing classical rational maps from Lattès-type geometries.

The Ahlfors regular conformal dimension is a fundamental quasi-symmetry invariant for compact metric spaces, groups, and dynamical systems. Defined as the infimum of the Hausdorff dimensions of all Ahlfors regular metrics within the quasisymmetric conformal gauge, it quantifies the minimal geometric dimension attainable while preserving controlled volumetric regularity under bounded distortion homeomorphisms. The invariant is computable via combinatorial modulus methods, and its attainment is linked to rigidity phenomena in conformal geometry, group theory, and dynamics. Central dichotomies emerge, especially in settings like coarse conformal dynamics on the 2-sphere and boundaries of hyperbolic groups, distinguishing classical rational and Lattès-type geometries.

1. Definition and Foundational Properties

Let $(X,d)$ be a compact metric space. A metric $d'$ on $X$ is called Ahlfors $Q$-regular if there exists a Borel measure $\mu$ and $C \geq 1$ such that, for all $x \in X$ and $0 < r \leq \operatorname{diam}(X)$,

$C^{-1} r^Q \leq \mu(B_{d'}(x, r)) \leq C r^Q.$

Given a homeomorphism $f : (X,d) \to (X,d')$, $f$ is quasisymmetric if there exists a homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that for all $x,y,z\in X$,

$$\frac{d'(f(x),f(y))}{d'(f(x),f(z))} \leq \eta\left(\frac{d(x,y)}{d(x,z)}\right).$$

The Ahlfors regular conformal gauge of $(X,d)$ is the set of all metrics $d'$ on $X$ that are quasisymmetric to $d$ and Ahlfors $Q$-regular for some $Q$. The Ahlfors regular conformal dimension is

$$\dim_{AR}(X,d) = \inf_{d' \in \mathcal{G}_{AR}(X,d)} \dim_H(X,d').$$

This infimum, when finite, only depends on the quasisymmetric class of $d$ and is always at least the topological dimension of $X$.

2. Computation via Combinatorial Modulus

The computation of Ahlfors regular conformal dimension (ARC-dim) is achieved via discrete combinatorial modulus formulations. Given a nested sequence of scale coverings or tree partitions, one considers the combinatorial $p$-modulus for families of discrete paths or curve systems. The critical exponent at which the $p$-modulus drops to zero is precisely the ARC-dim:

$\dim_{AR}(X,d) = \inf\{\,p>0 : M_p(X,d) = 0\,\}$

where $M_p(X,d)$ is the liminf of the supremum, over basepoints, of the discrete $p$-modulus for connecting paths in the hyperbolic filling structure or nerve graphs associated with the space. This discrete-to-continuum correspondence (the Carrasco Piaggio theorem) establishes the ARC-dim as the vanishing threshold for the combinatorial modulus of curve families or, equivalently, the critical exponent separating positive and vanishing $p$-energies of horizontal graphs in the corresponding tree or covering system (Esmayli et al., 2024, Piaggio, 2012, Kigami, 2018).

3. Attainment and Rigidity Phenomena

Attainment refers to the existence of a metric in the conformal gauge such that the Hausdorff dimension equals the ARC-dim. In self-similar and Laakso-type fractal spaces, explicit criteria for attainment are formulated via the non-vanishing of the optimal modulus density on the generator graph. The Ahlfors regular conformal dimension is attained if and only if no "removable edge" (one with zero density in the modulus minimizer) exists. The presence of such removable edges signals the existence of porous subsets of full ARC-dim, which preclude Ahlfors regular realization (Anttila et al., 20 Jan 2026).

For hyperbolic groups and dynamical systems, attainment is tightly linked to global geometric structure and dynamical rigidity. For topologically coarse expanding conformal (cxc) maps $f:S^2\to S^2$, a sharp dichotomy holds (Haïssinsky et al., 2011):

• If $Q=2$ and attained, $f$ is topologically conjugate to a semihyperbolic rational map.
• If $Q>2$ and attained, $f$ is topologically conjugate to a Lattès map arising from an expanding affine map of the torus with real, distinct eigenvalues.

For one-ended hyperbolic groups, ARC-dim is attained if and only if the boundary is a sphere or circle equipped with the standard conformal structure; otherwise, splittings and porosity of limit sets force non-attainment (Carrasco et al., 2020).

4. Connections to Conformal, Assouad, and Spectral Dimension

On quasiself-similar and combinatorially Loewner (CLP) spaces, the conformal Hausdorff, conformal Assouad, and Ahlfors regular conformal dimensions coincide: $\dim_{CH}(X) = \dim_{CA}(X) = \dim_{ARC}(X)$ This is established via a unified combinatorial modulus scheme that interpolates between classical modulus and energy formulations, providing a robust invariant for fractals such as the Sierpiński carpet (Eriksson-Bique, 2023). For infinite graphs and resistance-dominated Dirichlet spaces, ARC-dim is linked (often sharply) to spectral dimension and the asymptotic behavior of heat kernels (Sasaya, 2022, Sasaya, 2021, Sasaya, 2020). Typically,

$\dim_{AR}(X,d) \leq d_s < 2,$

but counterexamples arise in inhomogeneous settings.

5. Dynamics, Rational Maps, and Graph Energies

In complex dynamics, especially for post-critically finite rational maps $f$ with Julia set $\mathcal{J}_f$, the ARC-dim encapsulates the minimal geometric complexity permissible under the dynamics. The invariant is computed via energies of associated virtual endomorphisms of graphs extracted from subdivision rules or spines. The conformal dimension is characterized as the unique solution to the equation of asymptotic $q$-conformal energy equaling one: $$\overline{E}^{q_*}(f) = 1, \quad\text{with } \dim_{AR}(\mathcal{J}_f) = q_*,$$ where $\overline{E}^q$ is the asymptotic $q$-energy of the virtual endomorphism (Pilgrim et al., 2021, Park, 2022).

For Julia sets of crochet maps, ARC-dim is $1$ if and only if the map has an invariant graph of topological entropy zero, mirroring group-theoretic notions of Loewner property and cut-point spread (Park, 2022).

6. Connections to Poincaré Profiles and Group Theory

Ahlfors regular conformal dimension is directly linked to critical exponents of Poincaré profiles for hyperbolic cones and random groups. Explicitly, for a compact AR space $Z$, the critical Poincaré exponent of its cone $X$ satisfies

$p_\Lambda(X) = \dim_{AR}(Z),$

in a broad array of settings (product spaces, Heintze manifolds). This equality connects geometric analysis of Poincaré inequalities to the quasi-conformal geometry of boundaries and yields embedding obstructions: monotonicity of $p_\Lambda$ under regular maps precludes coarse embeddings between spaces of differing ARC-dim (Hume et al., 13 Nov 2025).

7. Stability, Semi-Continuity, and Open Problems

The Ahlfors regular conformal dimension is upper semi-continuous under Gromov-Hausdorff convergence in uniformly perfect, quasi-self-similar compact spaces, and is continuous for the boundaries of quasiconvex-cocompact groups in $\delta$-hyperbolic spaces under equivariant convergence (Cavallucci, 2022). ARC-dim is also realized as the critical exponent for vanishing combinatorial modulus in this limit process.

Outstanding problems include the necessity of quasiself-similarity or the CLP to ensure the equality of the various conformal dimension invariants, and the exact circumstances under which the spectral and Ahlfors regular conformal dimensions coincide, especially in non-self-similar geometries. The analytic relationships between ARC-dim, Poincaré/Separation profiles, and higher moduli invariants remain active research directions, as do explicit bounds for the quasisymmetry distortion realizing the minimal dimension.

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Key References: (Haïssinsky et al., 2011, Piaggio, 2012, Kigami, 2018, Esmayli et al., 2024, Anttila et al., 20 Jan 2026, Eriksson-Bique, 2023, Cavallucci, 2022, Carrasco et al., 2020, Park, 2022, Pilgrim et al., 2021, Hume et al., 13 Nov 2025, Sasaya, 2022, Sasaya, 2021, Sasaya, 2020)