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

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 $\mathrm{MG}$ of finitely generated marked groups. It is proved in this note that if a sequence ${\Gamma }_n\in \mathrm{MG}$ of marked groups converges to a group $\Gamma$, then $\overline{\lim }{\beta }_1({\Gamma }_n)\le {\beta }_1(\Gamma )$.