Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces

Abstract

The authors classify all topological types of degenerate central fibers appearing in holomorphic families of closed Riemann surfaces of genus g≥2 over the unit disc. A degenerating family of genus g is a triple (M,D,φ) consisting of a 2-dimensional complex manifold M, an open unit disk D in the complex plane, and a surjective proper holomorphic map φ such that all fibers of φ are connected and φ|φ−1(D∗): φ−1(D∗)→D∗ is a smooth fiber bundle with fiber Σg, where Σg is an oriented closed surface of genus g and D∗=D−{0}. The monodromy homeomorphism f: Σg→Σg of (M,D,φ) is determined as usual up to isotopy and conjugation. It is known that f is a pseudo-periodic homeomorphism of negative twist, that is, its mapping class [f] is either of finite order or reducible, and in the latter case, all component mapping classes are of finite order and its screw numbers are all negative. A family is said to be minimal if it is free of (−1)-curves. Two families (Mi,D,φi), i=1,2, are topologically equivalent if there exist homeomorphisms H:M1→M2 and h:D→D satisfying h(0)=0 and h∘φ1=φ2∘H. Let Sg={minimal degenerating families of genus g} modulo topological equivalence. Denote by P−g the set of all pseudo-periodic mapping classes of negative twist of Σg. Then we have a well-defined map monodromy ρ:Sg→P−g. The main result is the following theorem: For g≥2, ρ:Sg→P−g is bijective. The most essential part of the proof of this theorem is to construct the inverse map of ρ, that is, for a given pseudo-periodic homeomorphism f of negative twist the authors construct a degenerating family (M,D,φ) of genus g with monodromy homeomorphism f. In the second part of this paper the authors give a complete set of conjugacy invariants for the pseudo-periodic homeomorphisms of negative twist, which shows that Nielsen's set of invariants is not complete.