【Monopolize】 , 【engross】 , 【absorb】 , 【consume】 mean to take up completely.
【Monopolize】 , the general term, means to possess or control exclusively.
Occasionally 【engross】 implies getting a physical control of (as by purchase of the available supply) and this notion may persist in extended use, but more often the verb takes an immaterial object and implies a preoccupying.
【Absorb】 is frequently interchangeable with 【engross】 , but it is less often predicated of persons as conscious agents and more often of things that have an inherent capacity for monopolization.
【Consume】 comes into comparison with 【engross】 and 【absorb】 chiefly in an extended sense of each, implying monopolization of one’s time, attention, or interest.