In this series of papers we show that there are exactly ten subfactors, other
than A∞ subfactors, of index between 4 and 5. Previously this classification was known
up to index 3 +√3. In the first paper we give an analogue of Haagerup’s initial classification of subfactors of index less than 3 +√3, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.
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