Hyperelliptic graphs and the period mapping on outer space

Corey Bregman; Neil J. Fullarton

doi:10.1112/topo.12049

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arXiv · 1609.03597

Hyperelliptic graphs and the period mapping on outer space

Corey Bregman·Neil J. Fullarton

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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.

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Corey Bregman, Neil J. Fullarton. 2016-09-12. Hyperelliptic graphs and the period mapping on outer space. https://doi.org/10.1112/topo.12049

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