Editing Thompson groups (section)

==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]].