I was interested in this topic and found this post. I tried the mail to method suggested by @hobbesVT and it worked, but the Google groups service responds with a confirmation message that often lands in spam.

I am using a different approach not yet mentioned here.

https://groups.google.com/search?q=your+group+name

This will open the search page with your group as the top/only result. Then your prospective member can click on the link name and ask to join. It's still 2 clicks, but it cuts out a few steps and can be QR encoded.