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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.
 
I think you can use the iphone.facebook.com site, but that kind of kills the point of the app...
 
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