TITLE: Is there an infinite group with exactly two conjugacy classes?
QUESTION [16 upvotes]: Is there an infinite group with exactly two conjugacy classes?

REPLY [18 votes]: As a psychological curiosity, Per Enflo writes in his Autobiography that the existence of groups with two conjugacy classes was a key insight behind his many solutions to outstanding problems in Functional Analysis. 
"I made important progress in mathematics in 1966, but it was more on the level of new insights, than actual results. When thinking about topological groups in the spirit of Hilbert's fifth problem (I had gradually modified the Hibert problem to some very general program: To decide whether different classes of topological groups shared properties with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, where all elements except e, are conjugate to each other. I constructed such groups, by finding the right finite phenomenon and then make an induction. I understood, that this is a very general construction scheme (or "philosophy"), that probably could be applied to various infinite or infinite-dimensional problems. And actually – this philosophy is behind several of my best papers - the solution of the basis and approximation problem, the solution of the invariant subspace problem for Banach spaces, the solution of Smirnov's problem on uniform embeddings into Hilbert space and more."
( http://perenflo.com/sida9.html )