FIG. 2.

(a) The adjacency matrix can be approximated by the hyperbolic Laplacian in the continuum limit through Eq. (8). To derive this property, we choose an arbitrary site z_i with coordination number 3 (blue diamond). When applying the automorphism $z\mapsto w\left(z\right)=\frac{z_i-z}{1-z{z}_i}$, which maps z_i to the origin, the three neighbors of z_i (red squares) are mapped to an equilateral triangle. This implies Eq. (7), which can be expanded in powers of h to yield the desired relation. (b) Sums over lattice sites are replaced by integrals over hyperbolic space according to Eq. (9). This is achieved by assigning to each site z_i an effective hyperbolic triangle with interior angles 2π/7 and area ${\mathcal{A}}_{\Delta }=\pi /28$.