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.