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International Mathematics Research Notices Advance Access originally published online on June 23, 2009
International Mathematics Research Notices (2009) 2009:4336-4353, doi:10.1093/imrn/rnp090 published on October 27, 2009
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Nonunitarizable Representations and Random Forests

Inessa Epstein1,2 and Nicolas Monod1

1 Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

Correspondence: Correspondence to be sent to: nicolas.monod{at}epfl.ch

We establish a connection between Dixmier's unitarizability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is nonunitarizable if its first L2-Betti number is nonzero or if it is finitely generated with nontrivial cost. Our criterion also applies to torsion groups constructed by Osin, thus providing the first examples of nonunitarizable groups without free subgroups.

2 Present address: Caltech, Pasadena, CA 91125, USA.

Communicated by Prof. Marc Burger


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