W. A. Bogley, N. D. Gilbert, and James Howie, Cockcroft properties
of Thompson's group, Canad. Math. Bull. 43 (2000), 268-281.
Abstract
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 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.