Hyperelliptic graphs and the period mapping on outer space

TL;DR

This paper studies the period mapping on outer space related to free groups, analyzing fibers, hyperelliptic involutions, and the hyperelliptic Torelli group, revealing topological properties of hyperelliptic graph loci.

Contribution

It introduces the hyperelliptic Torelli group for free groups, analyzes the fibers of the period mapping, and shows topological simplifications when including degenerate graphs.

Findings

Fibers of the period mapping are aspherical and pi_1-injective.

Generators for the hyperelliptic Torelli group are obtained.

Adding degenerate graphs makes hyperelliptic loci simply-connected.

Abstract

The period mapping assigns to each rank n, marked metric graph Gamma a positive definite quadratic form on H_1(Gamma). This defines maps Phi* and Phi on Culler--Vogtmann's outer space CV_n, and its Torelli space quotient T_n, respectively. The map Phi 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 Phi in T_n, showing that they are aspherical, pi_1-injective subspaces. Metric graphs admitting a 'hyperelliptic involution' play an important role in the structure of Phi, leading us to define the hyperelliptic Torelli group, ST(n) < Out(F_n). We obtain generators for ST(n), and apply them to show that the connected components of the locus of 'hyperelliptic' graphs in T_n become simply-connected when certain degenerate graphs at infinity are added.