SQL 2014 Always On Availability Groups - Read Replica I'm not familiar with Always On Availability Groups, but our company is using it to provide a read-only replica for our main production database so that certain users can report against it and generate ad-hoc queries. 

Since this is read-only, I know that no modifications can be made to this database. However, I was wondering if it's ever necessary to rebuild indexes or regenerate statistics on a read replica like this? Or, does this happen automatically when the the source database has indexes rebuilt or statistics regenerated?
No, you don't need to rebuild indexes or update statistics. You can't do any writes to a readable replica at all.