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Cockcroft Properties of Thompson's Group

  Published:2000-09-01
 Printed: Sep 2000
  • W. A. Bogley
  • N. D. Gilbert
  • James Howie
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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 {\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.
Keywords: two-complex, covering space, Cockcroft two-complex, Thompson's group two-complex, covering space, Cockcroft two-complex, Thompson's group
MSC Classifications: 57M20, 20F38, 57M10, 20F34 show english descriptions Two-dimensional complexes
Other groups related to topology or analysis
Covering spaces
Fundamental groups and their automorphisms [See also 57M05, 57Sxx]
57M20 - Two-dimensional complexes
20F38 - Other groups related to topology or analysis
57M10 - Covering spaces
20F34 - Fundamental groups and their automorphisms [See also 57M05, 57Sxx]
 

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