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In 1979 Chr. Riedtmann introduced coverings of the Auslander-Reiten quiver F A of a representation-finite algebra A ([15]; see also 1.3 and 2.2 below). Our main results are that F A admits many finite coverings in general, and that each of these is the Auslander-Reiten quiver F M of some representation-finite M (2.9). In order to prove the first statement we show in w that the finite coverings of F A are classified by the actions of the fundamental group H (1.2) of F a on finite sets; in general, there are many such actions because H is a free (non-commutative) group (4.2). We obtain our second main statement by considering the algebra E which is defined by the mesh relations of a finite covering A of F A ([16], 1.4; see 2.5 below); such an E satisfies the Auslander conditions characterizing the algebras of the form End(@. M/), where M i ranges through chosen representatives of the indecomposable modules over some representation-finite algebra (2.3). In case A =Fa, the relations between E and A are studied in w 5.The theoretical notions developed in this paper give rise to concrete algorithms (and computer programs) which enable us to construct the AuslanderReiten quivers for plenty of algebras. We enter upon these algorithms in w tackling the special case H= 1. The general case will be examined in a subsequent publication, from which we borrow the Auslander-Reiten quivers of the 14 "maximal" algebras listed at the end of our paper (each basic connected representation-finite algebra with two simple modules is isomorphic to a quotient of a "maximal" algebra or to its opposite). The list of these maximal algebras has also been obtained by A.V. Nikulin and C.A. Panasiuk [14] as an application of the methods developed in Kiev.The present paper is intimately related to the results of Chr. Riedtmann ([15], [16]). Her unpublished collection of Auslander-Reiten quivers was a decisive help in proving that the fundamental group is free. Unfortunately, her own work on selfinjective algebras of class D, and the distance between Boston and Ziirich finally prevented us from carrying through the original plan of a common publication. We take pleasure in thanking her for encouragements and remarks.

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We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and +∞. Using these estimates we prove a spectral gap result by following the technique in [1], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.