I'm fairly new to Facebook, but I have the app.
Not sure what you mean by "events". But on the app, there is a home button, that you can see a news feed of everything that has been posted for you and you friends, plus a notifications tab, with all of those also. It's pretty much the same as you see on your page, only you can't edit much.
It's a nice app, works very well for me, even for posting pictures from the phone on my wall.
EDIT......
Ooops, I just figured out the "event" thing, you mean the calendar stuff. I don't see anything on my app that allows you to get to that.
Don't see a way to look at groups either.