Preprint: A simple description of Thompson's group F

Preprint: A simple description of Thompson's group F

[Tom Leinster]

The following paper is available:

"A simple description of Thompson's group F"

Marcelo Fiore, Tom Leinster

We show that Thompson's group F is the symmetry group of the "generic
idempotent".  That is, take the monoidal category freely generated by an
object A and an isomorphism A \otimes A --> A; then F is the group of
automorphisms of A.

http://arxiv.org/abs/math.GR/0508617


Incidentally, this is a result about groups, but the proof uses some
higher-dimensional category theory (multicategories, operads, and, less
essentially, bicategories).

Tom

Re: Preprint: A simple description of Thompson's group F

[Peter Freyd]

Marcelo and Tom write

  We show that Thompson's group F is the symmetry group of the "generic
  idempotent".  That is, take the monoidal category freely generated by an
  object A and an isomorphism A \otimes A --> A; then F is the group of
  automorphisms of A.

Tom has pointed out to me that the review of the old Freyd/Heller I
posted give no hint of its relevance. Therefor this:

  F was defined (40 years ago) as the initial model for a group with
  an endomorphism that's conjugate to its square.

More formally: consider the equational theory that adds to the theory
of groups a constant, s, and a unary operator  e, subject to two
further equations:

    e(xy) = (ex)(ey)             "e is a endomorphism"
    s(ex) = (e(ex))s             "e is a conjugacy-idempotent"

The initial algebra for this theory is the group  F.

(If one insists on removing the type-error in the last sentence, then
try "the initial algebra for this theory when subjected to the
forgetful functor back to groups is  F.")

If one defines a sequence of elements   s_n = e^n(s)  they clearly
generate  F (as a group) and it isn't hard to see that a complete set
of relations for  F (as a group) is the doubly-infinite family

     s_a s_b = s_{b+1} s_a

one such equation for each  a < b. (It took me ten years to find a
proof that just two of these equations imply all the others.)

Re: Preprint: A simple description of Thompson's group F

[Peter Freyd]

  There's a good chance that the characterization of Thompson's group F
  (not to mention its name) was set forth in the paper reviewed below
  (the authors of which became aware of R.J.Thompson's priority via
  this review).


Freyd, Peter; Heller, Alex
Splitting homotopy idempotents. II.
J. Pure Appl. Algebra 89 (1993), no. 1-2, 93--106.

A preliminary version of this paper was in the reviewer's hands in
1979 and was then of uncertain age. The authors have done a service in
publishing it (in somewhat revised form) belatedly.

The object of study is a free homotopy idempotent $f \colon X \to X$;
this means that $f$ is freely (base point not necessarily preserved
during the homotopy) homotopic to $f^2 \equiv f \circ f$. This $f$ is
said to split if there are maps $d \colon X \to Y$ and $u \colon Y \to
X$ such that $d \circ u \simeq \text{id}_Y$ and $u \circ d \simeq f$,
where $\simeq$ denotes free homotopy.

They construct a group $F$ and an endomorphism $\phi \colon F \to F$
such that, for a certain $\alpha_0 \in F$, $\phi^2(7) =
\alpha^{-1}_0\phi(7)\alpha_0$. The induced map $g \colon K(F,1) \to
K(F,1)$ is a homotopy idempotent which does not split; and it is
universal in the sense that it maps "canonically" into any homotopy
idempotent, and the corresponding homomorphism $F \to \pi_1(X)$ is
monic if and only if $f$ does not split.

This group $F$ is shown to be finitely presentable, has simple
commutator subgroup, is a totally ordered group and contains a copy of
its own infinite wreath-product. Every abelian subgroup is free
abelian, and every subgroup is either finite-rank free abelian or
contains an infinite-rank free abelian subgroup.

\{Reviewer's remarks: (1) While the authors acknowledge that some of
the above is due independently to J. Dydak
\ref[Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25
(1977), no. 1, 55--62; MR0442918 (56 \#1293)], they fail to mention
that priority for this group is generally given to R. J. Thompson
\ref[R. J. Thompson and R. McKenzie, in Word problems (Irvine, CA,
1969), 457--478, North-Holland, Amsterdam, 1973; MR0396769 (53
\#629)], who introduced $F$ and seemed to know many of its properties
in the late 1960s. Closely related to $F$ are Thompson's finitely
presented infinite simple groups. (2) Subsequently, as acknowledged by
the authors, much more became known about this extraordinary group. To
help the reader know what we are discussing, we mention that $F$ is
often known as "the Richard Thompson group"; also as the "Freyd-Heller
group", the "Dydak-Minc group" and (incorrectly, but because of later
work on $F$) as the "Brown-Geoghegan group". (3) The origin of the
curious name "$F$" was explained to the reviewer by one of the authors
as standing for "free", as in "free homotopy idempotent".\}

Reviewed by Ross Geoghegan