Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney--Sultan.

In this talk, we study connected components of Morse boundaries. 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 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.