On minimal shapes and topological invariants in hyperbolic lattices

We characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. The main tool is a layer decomposition due to Rietman--Nienhuis--Oitmaa and Moran, which allows us to prove convergence towards the Cheeger constant when these shapes exhaust the lattice. Furthermore, we apply a celebrated result of Floyd--Plotnick, which will allow us to compute the Euler characteristic for these graphs in terms of certain growth functions and the number of $n$-sized animals on those lattices.