Inventiones mathematicae

, Volume 151, Issue 1, pp 135-165

First online:

Some groups of type VF

  • Ian J. LearyAffiliated withFaculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, U.K. (e-mail:
  • , Brita E.A. NucinkisAffiliated withDepartement Mathematik, ETH Zürich, CH-8092 Zürich, Switzerland (e-mail:

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A group is of type VF if it has a finite-index subgroup which has a finite classifying space. We construct groups of type VF in which the centralizers of some elements of finite order are not of type VF and groups of type VF containing infinitely many conjugacy classes of finite subgroups. It follows that a group G of type VF need not admit a finite-type universal proper G-space. We construct groups G for which the minimal dimension of a universal proper G-space is strictly greater than the virtual cohomological dimension of G. Each of our groups embeds in GL m (ℤ) for sufficiently large m. Some applications to K-theory are also considered.