Group pages show a "Login to join group" link (opens in a new tab) if
you're not logged in.
If you're logged in they show a "Click to join group" button, posts to
new join() callable that adds the user to the group.
If you're already a member they show a "Share this link to invite people" link.

The groups dropdown list has new share icon links to the group's page,
for each group.