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 ${\partial }_{M}G$, we show that every connected component of ${\partial }_{M}G$ 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 Druţu-Sapir), we show that Morse boundaries of vertex groups are topologically embedded in ${\partial }_{M}G$.

16 pages; better title, final version; to appear in Pacific J. Math