Hyperelliptic graphs and the period mapping on outer space. Issue 1 (26th February 2018)

• Record Type: Journal Article
• Authors: Bregman, Corey; Fullarton, Neil J.
• Is Part Of: Journal of topology. Volume 11:Issue 1(2018)
• Page Start: 221
• Page End: 256
• Publication Date: 2018-02-26
• Subjects: 20F65 (primary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12049
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: The period mapping assigns to each rank n, marked metric graph Γ a positive definite quadratic form on H 1 ( Γ, R ) . This defines maps Φ ̂ and Φ on Culler–Vogtmann's outer space C V n, and its Torelli space quotient T n, respectively. The map Φ 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 Φ in T n, showing that they are aspherical, π 1 ‐injective subspaces. Metric graphs admitting a hyperelliptic involution play an important role in the structure of Φ, 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.