General:
* Renounce
* Relinquish
* Forgo
* Waive
* Surrender
* Abandon
* Give up
* Decline
* Reject
More formal:
* Disclaim
* Forego
* Abjure
* Recant
* Retire
Specific to power or authority:
* Devolve
* Delegate
* Yield
* Ceder
The best synonym for "abnegate" will depend on the specific context in which you are using it.