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

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