A variant of Hadamard’s notion of partie finie is applied to the theory of automorphic functions on arithmetic quotients of the upper half-plane. As a consequence, conceptually simple proofs of the volume formula and the Maass-Selberg relations are given. This technique interprets Zagier’s idea of renormalization (Jour. Fac. Sci. Univ. Tokyo 28 (1982), 415–437) so that it can be generalized eas...

We consider a discrete forest-fire model on the upper half-plane of the two-dimensional square lattice. Each site can have one of the following two states: “vacant” or “occupied by a tree”. At the starting time all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate 1, independently for all sites. If an occupied cluster reaches the boundary of the...