Hyperelliptic graphs and the period mapping on outer space

Abstract

The period mapping assigns to each rank n, marked metric graph Γ a positive definite quadratic form on H1(Γ,R). This defines maps (Formula presented.) and ф on Culler–Vogtmann's outer space CVn, and its Torelli space quotient ф, respectively. The map ф is a free group analog of the classical period mapping that sends a marked Riemann surface to its Jacobian. In this paper, we analyze the fibers of ф in Tn, showing that they are aspherical, π1-injective subspaces. Metric graphs admitting a hyperelliptic involution play an important role in the structure of ф, leading us to define the hyperelliptic Torelli group, ST(n) ≤ Out(Fn). We obtain generators for STn, and apply them to show that the connected components of the locus of ‘hyperelliptic’ graphs in Tn become simply connected when certain degenerate graphs at infinity are added.