Groups primarily attract spammers and many fake users and they can drive you nuts.
Here's a method to authenticate members and even monetize a Facebook group.
When someone request membership, instead of asking simple questions which can easily be lied on, require a valid email address and mobile phone number, and show a message that states periodical emails about the group and its activities will be sent. Send an immediate email message to which they must reply to be confirmed. This action will validate the email address.
Now you get primarily authentic members and secondly get to build a valid email and SMS database to which you can send periodic notices on the latest exciting group activities. The result is more member interaction and the opportunity to earn revenue from members who wish to place ads in the newsletters and text messages.
Charge nominal fees ($1 USD) for advertising posts. It's best to go for volume sales at low price than to get none because they price is high.