Cyber fraud

What exactly you mean by Public IP.  Any log in into any social media sites or any other web sites is protected by a Password created by you only.  Therefore, inform the Police when and from where you created the logins and normally logi-in.  What are the IP addresses from which the person who cheated the Girl logged in and the timings? Gather the details and proof of alibi about where were you at those times and produce to the Police. Better take the assistance of a Lawyer who is well versed in this kind of cases,  to deal with the Police on your behalf.

If you are innocent and have no such relations with such parties, Cyber Crime people cannot do any harm to you.   Someone might have used your ID, but from where it was used and through which gadget it has been used and your location etc., also are evidence to be considered by Police before holding you responsible.  If you are innocent you need not worry, but if you are over smart you can not escape.