On $L^1$-approximation of groups

A longstanding open problem in the intersection of group theory and operator algebras is whether all groups are MF, that is, approximated by asymptotic representations with respect to the operator norm. More generally, for $1\le p\le \mathrm{\infty }$, it has been asked by Thom in his ICM address whether there exist groups which are not approximated with respect to the Schatten $p$-norm. The cases of $1<p<\mathrm{\infty }$ were addressed in previous works. We settle the case $p=1$, solving a question left open by Lubotzky and Oppenheim.