We presume innocence around here. We don't arrest and ask questions later. The police, no doubt, will investigate. And if the story does not check out then they can arrest the man. But if not you can't just assume he lied, with no evidence.
And by the way once you have a gun and a man who knows you have slept with his wife, you have a very dangerous situation and you can assume your health or life is in jeopardy.
Fraud is also an angle here ~ take the wife, invite the guy over, shoot him in the garage three times (must have really been advancing ~ 3 shots?) ~ let's see what happens later on ~ this could be one smarmy mess yet.
Situations like this cry out for the use of water-boarding on somebody. We really need the truth.