Connected components of Morse boundaries of graphs of groups

Abstract

Let a finitely generated group $G$ split as a graph of groups. If edge groups are undistorted and do not contribute to the Morse boundary $\partial_MG$, we show that every connected component of $\partial_MG$ with at least two points originates from the Morse boundary of a vertex group. Under stronger assumptions on the edge groups (such as wideness in the sense of Dru\c{t}u-Sapir), we show that Morse boundaries of vertex groups are topologically embedded in $\partial_MG$.