Cockcroft Properties of Thompson's Group

http://dx.doi.org/10.4153/CMB-2000-034-8
Canad. Math. Bull. 43(2000), 268-281

  Published:2000-09-01
 Printed: Sep 2000

In a study of the word problem for groups, R.~J.~Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed that $F$ is the fundamental group of a finite two-complex $Z^2$ having Euler characteristic one and which is {\em Cockcroft}, in the sense that each map of the two-sphere into $Z^2$ is homologically trivial. We show that no proper covering complex of $Z^2$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft.