Obviously the big things that have already been mentioned.. but if they are on the registry, I would investigate it first, since while odds are that they did something serious, it is possible to get on the registry for public urination. But I would not tolerate sexual abuse of children or really any sexual abuse. And a phrase from a film comes to me, "If a man hits me, he only does it once..."
I see a lot of people saying cheating, but... lets just say I have differing views on relationships, where cheating likely couldn't occur.