Don't take this as authoritative, but I use the mobile interface most of the time, and I'm pretty certain there's no way to join a group.
If you can't get to a computer, go to the group page, scroll down to the bottom of the page and switch to the desktop view just long enough to hit the join button. Once you are joined, you can switch back to the mobile view and use the group as normal.
If anyone knows something I've missed, though, I'd love to hear it. I honestly think joining groups just got overlooked in the mobile interface. Things like that happen on a site this size.