Groups, Geometry, and Dynamics

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Volume 4, Issue 2, 2010, pp. 263–273
DOI: 10.4171/GGD/83

Published online: 2010-02-21

The Sigma invariants of Thompson’s group F

Robert Bieri[1], Ross Geoghegan[2] and Dessislava H. Kochloukova[3]

(1) Johann Wolfgang Goethe-Universität, Frankfurt am Main, Germany
(2) Binghamton University, USA
(3) IMECC - UNICAMP, Campinas, Brazil

Thompson’s group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute Σm(F) and Σm(F;ℤ), the homotopical and homological Bieri–Neumann–Strebel–Renz invariants of F, and show that Σm(F) = Σm(F;ℤ). As an application, we show that, for every m, F has subgroups of type Fm − 1 which are not of type FPm (thus certainly not of type Fm).

Keywords: Thompson’s group, finiteness properties, homological and homotopical Sigma invariants

Bieri Robert, Geoghegan Ross, Kochloukova Dessislava: The Sigma invariants of Thompson’s group F. Groups Geom. Dyn. 4 (2010), 263-273. doi: 10.4171/GGD/83