Semi-continuity of the first $\ell^2$-Betti number on the space of finitely generated groups

• Autores: Michael Pichot
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 81, Nº 3, 2006, págs. 643-652
• Idioma: inglés
• DOI: 10.4171/cmh/67
• Texto completo no disponible (Saber más ...)
• Resumen

  • To each finitely generated group is associated a sequence $(\beta_0, \beta_1,\beta_2,\dots)$ of non negative real numbers, its $\ell^2$-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 $\Gamma_n\in MG$ of marked groups converges to a group $\Gamma$, then $\overline\lim \beta_1(\Gamma_n)\leq \beta_1(\Gamma)$.