Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.