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The period mapping assigns to each rank   n  , marked metric graph   Γ   a positive definite quadratic form onH  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.

Hyperelliptic graphs and the period mapping on outer space