  We study connected components of the Morse boundary and their stabilisers. We
introduce the notion of point-convergence and show that if the set of
non-singleton connected components of the Morse boundary of a finitely
generated group $G$ is point-convergent, then every non-singleton connected
component is the (relative) Morse boundary of its stabiliser. The above
property only depends on the topology of the Morse boundary and hence is
invariant under quasi-isometry. This shows that the topology of the Morse
boundary not only carries algebraic information but can be used to detect
certain subgroups which in some sense are invariant under quasi-isometry.
