Dimer and Ising models on the Lobachevskii plane

The generating function for close-packet dimer configurations is studied for lattices constructed on the Lobachevskii plane using the Pfaffian method. These lattices are homogeneous under the modular group and the problem of counting dimer configurations is related to the word problem of Dehn. The partition function for the Ising model is found by solving a dimer problem using the prescription given by Fischer. The free energy is given as the solution of a set of algebraic equations and the specific heat has a power-law singularity with critical exponent $\alpha = 1$.