Help with an ex

My girlfriend’s ex is having a hard time letting go. He has been following us and watching to see whose car’s in her driveway. He told her when they broke up that he purposely took her to all of the restaurants in the area so that she would always think of him if she went with someone else. He’s also had his cop friend pull me over before work to get my information. He put his dog down because he knew how much she loved the dog, then he sent her a message that it was her fault. My question here is this: I think this is going to end very badly if something isn’t done but my girlfriend thinks if we ignore him he will eventually stop. What do we do?