Equivalence conjecture for infinite ideal polyhedra and circle patterns

Let $\mathcal{D}=(V,E,F)$ be a cellular decomposition of the sphere with dihedral angle $\Theta\in(0,\pi)^E$. An IIP or HIIP is an infinite or half-infinite ideal polyhedron, respectively; VEL-parabolic and VEL-hyperbolic refer to the corresponding vertex-extremal-length types, while recurrent and transient refer to the random walk on the 1-skeleton.

Equivalence conjecture for infinite ideal polyhedra. The following statements are equivalent, with the Euclidean alternatives applying to IIP and the hyperbolic alternatives applying to HIIP:

• (A) There exists an IIP (respectively, HIIP) with dihedral angle $\Theta$.
• (B) The 1-skeleton of $\mathcal{D}$ is VEL-parabolic (respectively, VEL-hyperbolic).
• (C) The 1-skeleton of $\mathcal{D}$ is recurrent (respectively, transient).
• (D) There exists a locally finite, embedded ideal circle pattern of $\mathcal{D}$ in $\mathbb{R}^2$ (respectively, the unit disk) with intersection angle $\Theta$.
• (E) The corresponding Ricci flow for ideal circle patterns in Euclidean background geometry (respectively, hyperbolic background geometry) converges.

The conjecture is motivated by work on infinite ideal circle patterns and related uniformization results. The source gives no resolution of this full equivalence, although several component existence and convergence theorems are established in the paper and related work.