Kähler groups, R -trees, and holomorphic families of Riemann surfaces

Thomas Delzant (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let X be a compact Kähler manifold, and g a fixed genus. Due to the work of Parshin and Arakelov, it is known that there are only a finite number of non isotrivial holomorphic families of Riemann surfaces of genus g 2 over X. We prove that this number only depends on the fundamental group of X. Our approach uses geometric group theory (limit groups, Ê−trees, the asymptotic geometry of the mapping class group), and Gromov-Shoen theory. We prove that in many important cases limit groups (in the sense of Sela) associated to infinite sequences of actions of a Kähler group on a Gromov-hyperbolic space are surface groups and we apply this result to monodromy groups acting on complexes of curves.

Domaines: Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

Thomas Delzant. Kähler groups, R -trees, and holomorphic families of Riemann surfaces. Geometric And Functional Analysis, 2016, ⟨10.1007/s00039-016-0355-x⟩. ⟨hal-01261997⟩