# Semi-continuity of the first $ℓ_{2}$-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 $(β_{0},β_{1},β_{2},…)$ of non negative real numbers, its $ℓ_{2}$-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 $Γ_{n}∈MG$ of marked groups converges to a group $Γ$, then $limβ_{1}(Γ_{n})≤β_{1}(Γ)$.

## Cite this article

Michael Pichot, Semi-continuity of the first $ℓ_{2}$-Betti number on the space of finitely generated groups. Comment. Math. Helv. 81 (2006), no. 3, pp. 643–652

DOI 10.4171/CMH/67