You have to be kidding. If they know who did it, they will pick him up immediately. The longer they leave Elizabeth in his hands, the better her chances of being killed. Unless they don't know where she is, and they're hoping he leads them to her.
As of today, though, I'm trying to stay away from speculation as to her whereabouts, or who did it. There are too many twists.
If they don't know where she is, but they do know who did it. They would absolutely grab the perp to find out where she is. Everyday they wait is a day closer to death for Elizabeth.