Is animal agriculture the leading cause of species extinction?

Question

The cause of species extinction interests me.

Cowspiracy: The Sustainability Secret is a documentary that attacks the environmental damage caused by animal agriculture.

Their facts page states (with references):

Animal agriculture is the leading cause of species extinction, ocean dead zones, water pollution, and habitat destruction. [xix] [iv]

The page also contains many statements that are outright misinterpretations (like the dead oceans by 2048, or 51% of all greenhouse emissions made by animal agriculture), and the references are beyond me, so I don't know if this is true.

Is animal agriculture the leading cause of species extinction, in the modern period?

Answer 1

No. According to Princeton University, Deforestation is the leading cause of species extinction.

Evidence to date suggests that deforestation is currently, and is projected to continue to be, the prime direct and indirect cause of reported extirpations. For example, it is predicted that up to 21% of Southeast Asian forest species will be lost by 2100 because of past and ongoing deforestation. Similar projections exist for biotas in other regions.[1]

Though "animal agriculture" isn't the leading cause, it may contribute to extinction of species if it amounts to Overexploitation.

Overexploitation is also an important driver of extinctions among vertebrates and tends to operate synergistically with other drivers such as habitat loss.[1]

Definition of Overexploitation:

Overexploitation means harvesting species from the wild at rates faster than natural populations can recover. Overfishing and overhunting are both types of overexploitation.[2]

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[1] http://press.princeton.edu/chapters/s5_8879.pdf

[2] http://www.shmoop.com/conservation-biology/threats-overexploitation.html