(From Tobira Gateway to Advanced Japanese, p.5.)
I thought 大きく is an adverb, so it can only qualify a verb; but 違い is a noun. What's going on here?
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This 違い is not a noun but a verb, and that's why it's modified by an adverb (大きく). This 違い is the continuative form (連用形) of 違う.
The continuative form of a verb has various usages. You can read a summary here: https://japanese.stackexchange.com/a/65953/5010 The second and the fifth usages in the last list are relevant to you now.