Journal of Operator Theory

Volume 66, Issue 1, Summer 2011 pp. 217-232.

New presentations of Thompson's groups and applications

Authors: Uffe Haagerup

$1$ and Gabriel Picioroaga

$2$
Author institution:

$1$ Department of Mathematical Sciences, University of Copenhagen, Universitetspark 5, 2100 Copenhagen O Denmark

$2$ Department of Mathematical Sciences, University of South Dakota, 414 E. Clark St., Vermillion SD 57069 U.S.A.

Summary: We find new presentations for the Thompson's groups

$F$ , the derived group

$F^{'}$ and the intermediate group

$D$ . These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group

$F^{'}$ is

$1$ hence the cost cannot decide the

$non$ amenability question of

$F$ ; the II

$_1$ factor

$L(F^{'})$ is inner asymptotically abelian and the reduced

$C^*$ -algebra of

$F$ is not residually finite dimensional.

Keywords: Thompson group, cost, von Neumann and

$C^*$ -algebras of a countable discrete group

Contents Full-Text PDF