With regard to your specific question can you sue LAPD for defamation for the dq based on alleged domestic abuse, I think it's debatable, but in the long run such a claim would fail.  First, there would need to be some kind of publication of the defaming comment/accusation - either a written publication or a verbal statement to a third party.  Ie., a letter addressed to you, or a statement made to you, informing you of the basis for the dq would not suffice.  Nor would internal communications among LAPD staff satisfy the publication requirement.  They would have to publicly "announce" in some manner infomration that was defamatory.  That's just the beginning - it doesn't even get into the problems of privileged communications, sovereign immunity, available remedial measures to appeal the decision, and so on.   

I'm a little unclear as to how their determination came up.  You were in the application process for 2.5 years?  Most LAPD tests are only valid for 1 year.  Did you have to retake them?  Also, how did the information about you yelling at your wife come to light?  A conviction for domestic battery?  Gossip from a neighbor?  Had you previously denied yelling at her, and they caught you in a lie?   

A little clarification would help.

Mongo