Pseudo-periodic Maps and Degeneration of Riemann Surfaces

Abstract

Surface mapping classes of algebraically finite type were introduced by Nielsen in 1944. Such a mapping class, in Thurston's classification, is reducible and its restriction to the reduced parts is of finite order. In the book under review, such a mapping class (or a homeomorphism in this class) is called pseudo-periodic. J. Nielsen studied these mapping classes extensively, in his famous paper [Danske Vid. Selsk. Math.-Phys. Medd. 21 (1944), no. 2, 89 pp.;]. He introduced invariants of these classes and of maps that represent them, like the screw number that measures the amount of Dehn twist performed by a power of the map on an invariant curve, and he also introduced an invariant called the `amphidrome'. He asserted that the invariants he introduced are complete conjugacy invariants for pseudo-periodic maps. Nielsen's arguments are sometimes considered as too vague, and one of the main objects of this book is to make Nielsen's assertions precise. For this purpose, the authors add to Nielsen's invariants a surface homeomorphism invariant called its `action on the partition graph'. The book under review is divided into two parts. The main result of Part I is to prove that Nielsen's invariants, together with the action on the partition graph, are complete invariants of the conjugacy class of a pseudo-periodic map. Part II contains applications of the results of Part I to the degeneration theory of Riemann surfaces. The main result here says that there is a bijective correspondence between topological types of minimal degenerating families of closed connected oriented Riemann surfaces of genus ≥2 over a disk which are nonsingular outside the origin and the conjugacy classes of mapping classes of the surface that are represented by mapping classes of negative Dehn twist. The correspondence is given by the topological monodromy. The results in this book are given with detailed proofs. The book is well written and it should be useful for low-dimensional topologists and algebraic geometers.