JournalscmhVol. 81 , No. 3DOI 10.4171/cmh/67

Semi-continuity of the first <em><em>l</em><sup>2</sup></em>-Betti number on the space of finitely generated groups

  • Michael Pichot

    École Normale Supérieure de Lyon, France
Semi-continuity of the first <em><em>l</em><sup>2</sup></em>-Betti number on the space of finitely generated groups cover

Abstract

To each finitely generated group is associated a sequence (β0,β1,β2,)(\beta_0, \beta_1,\beta_2,\dots) of non negative real numbers, its 2\ell^2-Betti numbers. On the other hand, the set of finitely generated groups desingularizes into a usual topological space, the space MG of finitely generated marked groups. It is proved in this note that if a sequence ΓnMG\Gamma_n\in MG of marked groups converges to a group Γ\Gamma, then limβ1(Γn)β1(Γ)\overline\lim \beta_1(\Gamma_n)\leq \beta_1(\Gamma).