Editing Thompson groups

{{Short description|Three groups}}
{{About| the three unusual infinite groups  F, T and V found by Richard Thompson| the sporadic simple group found by John G. Thompson | Thompson sporadic group}}

In [[mathematics]], the '''Thompson groups''' (also called '''Thompson's groups''', '''vagabond groups''' or '''chameleon groups''') are three [[group (mathematics)|groups]], commonly denoted <math>F \subseteq T \subseteq V</math>, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the [[von Neumann conjecture]]. Of the three, ''F'' is the most widely studied, and is sometimes referred to as '''the Thompson group''' or '''Thompson's group'''.

The Thompson groups, and ''F'' in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but [[finitely presented group|finitely presented]]. The groups ''T'' and ''V'' are (rare) examples of infinite but finitely-presented [[simple group]]s. The group ''F'' is not simple but its [[derived subgroup]] [''F'',''F''] is and the quotient of ''F'' by its derived subgroup is the [[free abelian group]] of rank 2.  ''F'' is [[totally ordered group|totally ordered]], has [[growth rate (group theory)|exponential growth]], and does not contain a [[subgroup]] isomorphic to the [[free group]] of rank 2.

It is conjectured that ''F'' is not [[amenable group|amenable]] and hence a further counterexample to the long-standing but recently disproved
[[von Neumann conjecture]] for finitely-presented groups: it is known that ''F'' is not [[elementary amenable]].

{{harvtxt|Higman|1974}} introduced an infinite family of finitely presented simple groups, including Thompson's group ''V'' as a special case.

==Presentations==

A finite [[presentation of a group|presentation]] of ''F'' is given by the following expression:

:<math>\langle A,B \mid\ [AB^{-1},A^{-1}BA] = [AB^{-1},A^{-2}BA^{2}] = \mathrm{id} \rangle</math>

where [''x'',''y''] is the usual group theory [[Commutator#Group theory|commutator]], ''xyx''<sup>−1</sup>''y''<sup>−1</sup>.

Although ''F'' has a finite presentation with 2 generators and 2 relations,
it is most easily and intuitively described by the infinite presentation:

:<math>\langle x_0, x_1, x_2, \dots\ \mid\ x_k^{-1} x_n x_k = x_{n+1}\ \mathrm{for}\ k<n \rangle.</math>

The two presentations are related by ''x''<sub>0</sub>=''A'', ''x''<sub>''n''</sub> = ''A''<sup>1&minus;''n''</sup>''BA''<sup>''n''&minus;1</sup> for ''n''>0.

==Other representations==

[[File:AA Tree Skew2.svg|right|thumb|The Thompson group ''F'' is generated by operations like this on  binary trees. Here ''L'' and ''T'' are nodes, but ''A'' ''B'' and ''R'' can be replaced by more general trees.]]
The group ''F'' also has realizations in terms of operations on ordered rooted [[binary tree]]s, and as a subgroup of the piecewise linear [[homeomorphism]]s of the [[unit interval]] that preserve orientation and whose non-differentiable points are [[dyadic rational]]s and whose slopes are all powers of 2.

The group ''F'' can also be considered as acting on the [[unit circle]] by identifying the two endpoints of the unit interval, and the group ''T'' is then the group of automorphisms of the unit circle obtained by adding the homeomorphism ''x''→''x''+1/2 mod 1 to ''F''. On binary trees this corresponds to exchanging the two trees below the root.  The group ''V'' is obtained from ''T'' by adding the discontinuous map that fixes the points of the half-open interval [0,1/2) and exchanges [1/2,3/4) and [3/4,1) in the obvious way. On binary trees this corresponds to exchanging the two trees below the right-hand descendant of the root (if it exists).

The Thompson group ''F'' is the group of order-preserving automorphisms of the free [[Jónsson–Tarski algebra]] on one generator.

==Amenability==
The conjecture of Thompson that ''F'' is not [[amenable group|amenable]] was further popularized by R. Geoghegan—see also the  Cannon–Floyd–Parry article cited in the references below. Its current status is open: E. Shavgulidze<ref>{{citation | last=Shavgulidze | first=E. | title=The Thompson group F is amenable 
| mr=2541392 | year=2009 | journal= Infinite Dimensional Analysis, Quantum Probability and Related Topics|volume=12  | issue=2 | pages=173–191 | doi=10.1142/s0219025709003719}}</ref> published a paper in 2009 in which he claimed to prove that ''F'' is amenable, but an error was found, as is explained in the [[Mathematical Reviews|MR]] review.

It is known that ''F'' is not [[elementary amenable]], see Theorem 4.10 in Cannon–Floyd–Parry. 

If ''F'' is '''not'''  amenable, then it would be another counterexample to the now disproved [[von Neumann conjecture]] for finitely-presented groups, which states that a finitely-presented group is amenable [[if and only if]] it does not contain a copy of the free group of rank 2.

==Connections with topology==

The group ''F''  was rediscovered at least twice by topologists during the 1970s. In a paper that was only published much later but was in circulation as a preprint at that time, [[Peter Freyd|P. Freyd]] and A. Heller <ref>{{citation | last1=Freyd | first1=Peter | last2=Heller | first2=Alex |  title=Splitting homotopy idempotents | mr=1239554 | journal= [[Journal of Pure and Applied Algebra]]|volume=89 | issue=1–2 |year=1993 |pages=93–106 | doi=10.1016/0022-4049(93)90088-b| doi-access=free }}</ref> showed that the ''shift map'' on ''F'' induces an unsplittable [[homotopy]] idempotent on the [[Eilenberg–MacLane space]] ''K(F,1)'' and that this is universal in an interesting sense. This is explained in detail in Geoghegan's book (see references below). Independently, J. Dydak and P. Minc <ref>{{citation | last1=Dydak | first1=Jerzy | last2=Minc | first2=Piotr |  title=A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR's | mr=0442918 | journal=Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques |volume=25 |year=1977 |pages=55–62}}</ref> created a less well-known model of  ''F'' in connection with a problem in shape theory.

In 1979, R. Geoghegan made four conjectures about ''F'': (1) ''F'' has [[finiteness properties of groups|type '''FP'''<sub>∞</sub>]]; (2) All homotopy groups of ''F'' at infinity are trivial; (3) ''F'' has no non-abelian free subgroups; (4) ''F'' is non-amenable. (1) was proved by K. S. Brown and R. Geoghegan in a strong form: there is a K(F,1) with two cells in each positive dimension.<ref>{{citation | last1=Brown | first1=K.S.| last2=Geoghegan | first2=Ross |  title=An infinite-dimensional torsion-free FP_infinity group | mr=0752825 |volume=77 |year=1984 |pages=367–381 | doi=10.1007/bf01388451|bibcode=1984InMat..77..367B }}</ref> (2) was also proved by Brown and Geoghegan <ref>{{citation | last1=Brown | first1=K.S.| last2=Geoghegan | first2=Ross |  title=Cohomology with free coefficients of the fundamental group of a graph of groups | mr=0787660 | journal= [[Commentarii Mathematici Helvetici]]|volume=60 |year=1985 |pages=31–45 | doi=10.1007/bf02567398}}</ref> in the sense that the cohomology H*(F,ZF) was shown to be trivial; since a previous theorem of M. Mihalik <ref>{{citation | last=Mihalik | first=M. | title=Ends of groups with the integers as quotient 
| mr=0777262 | journal= Journal of Pure and Applied Algebra|volume=35 |year=1985 |pages=305–320 | doi=10.1016/0022-4049(85)90048-9| doi-access= }}</ref> implies that ''F'' is simply connected at infinity, and the stated result implies that all homology at infinity vanishes, the claim about homotopy groups follows. (3) was proved by M. Brin and C. Squier.<ref>{{citation | last1=Brin | first1=Matthew. | last2=Squier | first2=Craig |  title=Groups of piecewise linear homeomorphisms of the real line | mr=0782231 | journal= [[Inventiones Mathematicae]]|volume=79 | issue=3 |year=1985 |pages=485–498 | doi=10.1007/bf01388519|bibcode=1985InMat..79..485B }}</ref> The status of (4) is discussed above.

It is unknown if ''F'' satisfies the [[Farrell–Jones conjecture]]. It is even unknown if the Whitehead group of ''F'' (see [[Whitehead torsion]]) or the projective class group of ''F'' (see [[Wall's finiteness obstruction]]) is trivial, though it easily shown that ''F'' satisfies the strong Bass conjecture.

D. Farley <ref>{{citation | last=Farley | first=D. | title=Finiteness and CAT(0) properties of diagram groups | mr=1978047 | journal=[[Topology (journal)|Topology]] |volume=42 | issue=5 |year=2003 |pages=1065–1082 | doi=10.1016/s0040-9383(02)00029-0| doi-access= }}</ref> has shown that ''F'' acts as deck transformations on a locally finite CAT(0) [[cubical complex]] (necessarily of infinite dimension). A consequence is that ''F'' satisfies the [[Baum–Connes conjecture]].

==See also==

*[[Higman group]]
*[[Non-commutative cryptography]]

==References==

{{reflist}}
*{{Citation | last1=Cannon | first1=J. W. | author-link1 = James W. Cannon | last2=Floyd | first2=W. J. | author-link2 = William Floyd (mathematician) | last3=Parry | first3=W. R. | title=Introductory notes on Richard Thompson's groups | url=http://www.math.binghamton.edu/matt/thompson/cfp.pdf | mr=1426438 | year=1996 | journal=L'Enseignement Mathématique |series=IIe Série | issn=0013-8584 |volume=42 | issue=3 | pages=215–256}}
*{{cite journal
| last1 = Cannon
| first1 = J.W.
| last2 = Floyd
| first2 = W.J.
| title = WHAT IS...Thompson's Group?
| journal = [[Notices of the American Mathematical Society]]
| issn = 0002-9920
|date=September 2011
| volume = 58
| issue = 8
| pages = 1112–1113
| url = http://www.ams.org/notices/201108/rtx110801112p.pdf
| access-date = December 27, 2011}}
* {{Citation | first=Ross | last1=Geoghegan | title=Topological Methods in Group Theory 
| publisher=[[Springer Verlag]] | year=2008 | isbn=978-0-387-74611-1 | series=[[Graduate Texts in Mathematics]] | mr=2325352 | volume=243 | doi=10.1142/S0129167X07004072| arxiv=math/0601683 }}
*{{Citation | last1=Higman | first1=Graham | author1-link=Graham Higman | title=Finitely presented infinite simple groups | url=https://books.google.com/books?id=LPvuAAAAMAAJ | publisher=Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra | series=Notes on Pure Mathematics | isbn=978-0-7081-0300-5 |mr=0376874 | year=1974 | volume=8}}

[[Category:Infinite group theory]]