# Why are Thompson's groups called $F$, $T$ and $V$?

Why are Thompson's groups called $$F$$, $$T$$ and $$V$$?

I never saw Thompson's unpublished notes, in which he introduces these groups; maybe an explanation can be found there?

• The groups I am referring to are those described here: Introductory notes on Richard Thompson's groups. May 11, 2019 at 10:40
• Note: there are several Richard Thompson referred in MathSciNet; the concerned one is Richard J. Thompson.
– YCor
May 11, 2019 at 11:12
• I think that $V$ was initially denoted as $G$ (it's the case in Higman's 1974 notes, where he denotes $V_{n,r}$ the free $n$-Jónsson-Tarski algebra on $r$ generators and $G_{n,r}$ its automorphism group, known as Thompson-Higman group now).
– YCor
May 11, 2019 at 12:15
• Looking at the notes, where past naming is discussed, I am guessing $F$ came from finiteness properties ($FP_\infty$) and for some reason it stuck
– user35370
May 11, 2019 at 16:37

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 (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it wasn't being used, or something (actually, according to Cannon–Floyd–Parry, Thompson called it $$\hat{V}$$ in his unpublished handwritten notes, so maybe that's part of the reason).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!)

• welcome to MathOverflow, professor Zaremsky! by the way, you might consider registering so that this "user 164670" ID becomes more informative... Sep 2, 2020 at 13:31
• On the other hand you might prefer to retain a mild degree of anonymity, because my experience is that if you are easily contactable then you will be contacted with lots of questions, requests for help, etc. Sep 2, 2020 at 14:23
• I added a link to Freyd–Heller, and, since e-Periodica seems to be down and I couldn't find it through MathSciNet, copied @AGenevois's link to Cannon–Floyd–Parry, but I couldn't figure out which of the many many many articles by Dydak was meant. Maybe you could clarify? Sep 3, 2020 at 12:58
• The Dydak paper is called, "A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR’s." And yeah, registering is probably smart, I just figured out how! Sep 4, 2020 at 10:50
• Concerning the names in Thompson's original notes: $F$ is called $\mathbb{P}$ as given by its finite presentation and $\hat{\mathbb{P}}$ as generated by piecewise linear homeomorphisms (so $\mathbb{P} \cong \hat{\mathbb{P}}$; $T$ is called $C$; and $V$ is indeed called $V$ (and there are hat versions of both of these as well). Feb 23, 2022 at 21:08

Matt Zaremsky's justification of "$$F$$" for "free" can also be found in Ross Geoghegan's review MR1239554 on Mathscinet.

I also took a look at McKenzie and Thompson's article An elementary construction of unsolvable word problems in group theory (MSN) (which seems to be the first publication mentioning the groups constructed by Thompson), and there is some history in §8. Below is a screenshot:

The groups $$\mathfrak{P}$$ and $$\mathfrak{P}'$$ correspond to the group now called $$F$$. From the context, $$\mathfrak{P}$$ probably refers to the "p" of "permutation". The group $$\mathfrak{C}'$$ corresponds to the group now called $$V$$. I would guess that $$\mathfrak{C}$$ refers to the "C" of "Cantor space".

• Concerning the names $C$ vs. $V$ Thompson writes in the notes I have "(Let $\hat{V}$ be the $\mathcal{C}'$ of p. 475 of Word Problems adjusted to the reals\ldots)" which raises questions of chronology I cannot resolve. Feb 24, 2022 at 6:17