There are three types of Facebook Group, public, closed and secret.
The Housing Alternatives Facebook Group has a security status of Closed. If it was public we would get more trolls than we are already getting, and if it was secret we would get no new members, so Closed is the best we can do, to both attract new members and limit trolls, but it has its limitations.
What non-members can and cannot see
It is important to understand what non-members can and cannot see. The Closed status is NOT anonymous or invisible.
Non-members CAN see the membership list. That means anyone who wants their participation in any Closed group secret must sign up to Facebook with a false identity. There is no way around this.
In the normal way no-one from outside the group can see posts to the group. Posts from a Closed group cannot be shared using the share button. If you want to share an attached article to a post, you have to go into that article, and share from the original.
But there are two notable and potentially dangerous exceptions.
If a member of a group decides to betray the group’s confidentiality, they can copy the link to a Post, and paste that link into another group. The post will appear in the other group, with the name of the person who has made the Post, the comments they have made in the status and any attachment connected to the post. Only group members can do this. I only discovered this possibility myself because one of my posts was shared on a group I was a member of, so I got to see the share. If I had not been a member of that group I would not have seen it or known about the group member breaching our confidentiality. That member of our group has been summarily removed and blocked.
In addition, anyone can take a photo any time, of posts, comments or anything they like, and post it anywhere they like.
To my knowledge, there is no way that comments on posts can be shared, but no guarantees. Facebook changes the way it works on pretty much a daily basis, and it is impossible for Admins to keep up. And as previously mentioned, anyone can take a photo any time, of posts, comments or anything they like, and post it anywhere they like.
This all means that you MUST assume your own responsibility for your own on-line security when using Facebook. Facebook is riddled with holes and anomalies.
Create a second identity
The best way to keep your personal identity anonymous or invisible is to create a false identity that you use for anything that you do not want stalkers or employers to see. This is technically against Facebook policy but you won’t get caught if you use a sensible name and start to connect that identity with others. IT boffins can still breach this, but random people are unlikely to go to the effort.
If you do create a new false identity be aware that admins refuse membership to new and empty profiles, because they are likely to belong to trolls. So you will have to notify the admin of the reason why your profile is new and empty before they will grant you membership of the group.
Over time you can build some substance to your new identity. Joining groups with a consistent theme, in a consistent location is one good way. You can also connect to new friends, who are also trying to protect their privacy from stalkers or employers or landlords or… even family members.
When I look through the membership list of Housing Alternatives I can see quite a few obviously false identities. But I would bet there are many more that do not hit the casual viewer as false. Go for it. It is the only way to protect your identity on Facebook.