The answer is yes, but the amount of work varies.
By calling
var user = Discourse.User.current();
You can get access immediately to their primary_group_name, but it won’t provide you with all groups they have access to.
To get all groups, you might have to do a call to /u/username.json and read the groups property. That makes it a tad more difficult as now you are awaiting the response of a call before you can show the link.