If you go offroad I know there is a Tuesday trail group, look for "Tirsdag" on this forum:
http://terrengsykkel.no/ubb/ubbthreads.php?ubb=postlist&Board=18&page=1SK Rye also has a usual 50km MTB gravel route they ride every Tuesday, you just meet up at 1745 at Brekkekrysset in Maridalen and ask to join!
However, for road cycling there is insofar as I know a lack of regular group rides (or maybe I'm just not cool enough to know of them!).
I usually just ask some of my cycling friends if they want to go out, but most of the time I ride by myself.
I'd be happy to take you along and show you some nice routes though! I live in the city center and go for both short and long rides in the afternoons or daytime in weekends, if you text me at 95231950 we can meet up.
As for Rye:
The mid-level Rye road cycling teams (there are a lot of them) only focus on Styrkeprøven, the (team) race from Trondheim to Oslo, as well as the shorter Lillehammer-Oslo distance of the race. All of this happens early July, so from April to July you shouldn't expect to be allowed to join in the regular group rides unless you have actually joined the team and aim to complete the Styrkeprøven with them, or if the captain has OK-ed it. This is for security reasons, we don't want any strangers in the paceline who aren't familiar with our quirks and norms since that could easily cause dangerous situations.
So for Rye in early season it's in or out, basically

After July though, things relax and the teams may train for other races, or just ride for fun, but I'd advise talking to one of the team captains first. Sunday rides are usually very easy-going.
You can see SK Ryes activity calendar here:
http://www.sportsklubbenrye.no/Road teams Rye15-17-19 etc are the full distance teams with the number denoting the expected race time.
Teams with Lille33-Lille37/Nivå3-Lille40 are the Lillehammer-Oslo teams with the number denoting the expected average speed.