Every finitely generated group \(G\) has an associated topological space, called a Morse boundary. It was introduced by a combination of Charney--Sultan and Cordes and captures the hyperbolic-like behavior of \(G\) at infinity.
We introduce the notion of point-convergence and show that if the set of non-singleton connected components of the Morse boundary of a 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 can be used to detect certain subgroups which in some sense are invariant under quasi-isometry. This is joint work with Bobby Miraftab and Stefanie Zbinden