After a quick tally it looks like the winner of the first possibly-semi-regular “name the mathematical opject” competition is ambiently trivial subgroup, which was suggested by Henry Wilton. Congratulations, Henry, and remember: you can still add this to your CV on MathJobs.org (though I would advise against it.) I was sort of hoping wrapping class group would win, but that’s because I’m a sucker for puns. (Is “wrapping class group” a pun?) The close runner up (if you don’t throw out Saul’s vote for himself) was Isot(H, M) (or Isot(M, \Sigma) as I would probably write it) which, Dave Futer pointed out is a notation rather than a name, so we’ll see if that catches on. (Something like Amb(M, \Sigma) might be easier to remember given the name of the subgroup.) Interestingly, there were more candidates than votes cast.
If you have any ideas for objects that you would like to see named by the second possibly-semi-regular “name the mathematical object” contest, feel free to suggest it in the comments, or by sending me an e-mail.