Can you please give your feedback regarding this scenario :

Customer Active directory domain name is : domain. Local

Two UPN are using while logging in Active Directory :

1) alpha.com

2) beta.com

While deploying ADFS they  don’t want to use  any single company domain name in ADFS Federation name just like “fs.alpha.com” or fs.beta.com”. There requirement is to register a new domain on internet and want to use a third name for ADFS Federation name such as “fs.ourgroup-mail.com”. Is this works or I just only bind to use any upn name in my ADFS Federation name.