What is the ruling regarding a man saying to his wife: "When Friday comes, I will tell you, 'You are divorced'," and then Friday comes and he says nothing? Is it a promise of divorce or an actual divorce? And does saying, "You are thrice divorced" in jest constitute a divorce?

This is a promise of divorce, not an enactment of it, so the wife is not considered divorced by it. The phrase "طالع" (tali') is neither an explicit divorce nor a metaphorical one. One should avoid such dangerous language. Some scholars consider divorce to be enacted by any word intended for divorce.