First of all: all the answers may apply to the management of many different types of digital communities, especially those with limited user verification options or sensitive subject matter, but this question pertains specifically to the management of Facebook groups. That may not be the typical community management scenario discussed here, but in my country the majority of Internet users now prefer to join social media site groups as opposed to forum sites. Difference in digital culture, I suppose.
The problem: I am currently the sole active admin of a couple of Facebook groups bringing together and providing a digital "safe space" to socially persecuted and marginalized groups of people. As such, it is of utmost importance to screen all potential new members to protect existing and future members from further harassment, trolling, as well as cyber vigilantes and the possibly life-threatening consequences they pose. At the same time (unsurprisingly) a number of legitimate applicants may and occasionally do request to join using assumed/"fake" identities as a fail-safe, so a fake profile may not necessarily be grounds to dismiss a request. Compounding this is the need to grow the group and reach out to more people in need of a supportive community they may not have access to in their immediate surroundings.
The question: What methods and policies may be implemented to effectively screen new members for a group, especially in such situations? Facebook already verifies its users in two major ways: valid email; age of account. The former is easy to fake and the latter may be misleading. (I'm not sure of the phone verification requirement--it doesn't seem to be implemented with consistency, even though it can also be faked; as for their "real name" policy, so far it only seems to have flagged those of my friends actually using their real names. Perhaps they haven't yet updated whatever database that system relies upon to reflect their multicultural user base.)