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> Why are [Thompson's groups](https://en.wikipedia.org/wiki/Thompson_groups) called $F$, $T$ and $V$?
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I never saw Thompson's unpublished notes, in which he introduces these groups; maybe an explanation can be found there?



Why are Thompson's groups called $F$, $T$ and $V$?
The "$F$" actually stands for "free homotopy idempotent", since $F$ is the universal group encoding a free homotopy idempotent (the endomorphism sending each standard generator $x\_i$ to $x\_{i+1}$ is idempotent up to conjugation). The universality was proved by [Freyd–Heller](https://doi.org/10.1016/0022-4049(93)90088-B) (who called it "$F$" for this reason), and independently Dydak. The group was subsequently sometimes called the "Freyd–Heller group", giving the notation $F$ a double meaning. Then people realized Thompson had worked with the group before Freyd–Heller did, and started calling it "Thompson's group $F$". The "$T$" just stands for "Thompson". The "$V$" is a little mysterious; for a while it was denoted "$G$" but I think people just realized "G" was bad notation for a specific group, and somehow "$V$" emerged as a good option since it wasn't being used, or something (actually, according to [Cannon–Floyd–Parry](http://people.math.binghamton.edu/matt/thompson/cfp.pdf), Thompson called it $\hat{V}$ in his unpublished handwritten notes, so maybe that's part of the reason).


(P.S. This is Matt Zaremsky, I don't really use MathOverflow but I thought I should answer this question!)


