# 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

## Abstract

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 $\overline\lim \beta_1(\Gamma_n)\leq \beta_1(\Gamma)$.

## Cite this article

Michael Pichot, Semi-continuity of the first <em><em>l</em><sup>2</sup></em>-Betti number on the space of finitely generated groups. Comment. Math. Helv. 81 (2006), no. 3, pp. 643–652

DOI 10.4171/CMH/67