  Phylogenetic networks represent evolutionary histories of sets of taxa where
horizontal evolution or hybridization has occurred. Placing a Markov model of
evolution on a phylogenetic network gives a model that is particularly amenable
to algebraic study by representing it as an algebraic variety. In this paper,
we give a formula for the dimension of the variety corresponding to a
triangle-free level-1 phylogenetic network under a group-based evolutionary
model. On our way to this, we give a dimension formula for codimension zero
toric fiber products. We conclude by illustrating applications to
identifiability.
