To my understanding this is a one-to-many relationship (a single account with a single value can appear many times in the Groups-Accounts dimension), but I'm not sure if: a) this can be handled as such in SSAS without issues and performance penalties, and b) it's possible to convert this to a regular relationship to keep a star schema, which would be the ideal solution.
The answer to these is yes. There can be a performance hit if the names can't be designated as unique, but it's normally not signficant. My location has a lot of repeated city and the county names which repeat in the various states. Despite having one entry per zip code, there is no human noticeable delay retrieving the data.
I need to see all the relevant columns in the dimension table. You show them in a heirarchy. That means there should be at least three columns in the dimension: surrogate ID, group, account. The latter two could even have their own key, which I would find ideal.
Thanks,