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Each of Thompson's groups F, T, and V has infinitely many ends relative to the groups F [0, 1/2], T [0, 1/2], and V [0, 1/2) (respectively). We can therefore simplify the proof, due to Napier and Ramachandran, that F, T, and V are not Kähler groups. Thompson's groups T and V have Serre's property FA. The original proof of this fact was due to Ken Brown.

A proof that Thompson's groups have infinitely many relative ends