1. to give a share ("abgeben von" - usually for small scale things like sharing a piece of your pizza)
2. to hand over, to turn in (For items that you "turn in", like a phone at the door of a yoga retreat. Also for some borrowed things like a bike or a car. And for roles and responsibilities.)
3. to be, to make for ("etwas+Acc abgeben" - Colloquial phrase for objects and people doing a surprisingly good job in a function that they weren't made for.)
4. to deal with, to occupy oneself with ("sich+Acc abgeben mit" - Usually used in the negative, either for tasks you don't want to deal with or people you (don't) want to get involved with.)
abgeben