Quantum isometries and group dual subgroups

We study the discrete groups $\mathrm{\Lambda }$ whose duals embed into a given compact quantum group, $\widehat{\mathrm{\Lambda }}\subset G$. In the matrix case $G\subset U_n^{+}$ the embedding condition is equivalent to having a quotient map ${\mathrm{\Gamma }}_U\to \mathrm{\Lambda }$, where $F=\{{\mathrm{\Gamma }}_U\mathrm{\mid }U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon’s classification of group dual subgroups $\widehat{\mathrm{\Lambda }}\subset S_n^{+}$. These results are motivated by Goswami’s notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.