Connected components of Morse boundaries of graphs of groups

TL;DR

This paper investigates the structure of Morse boundaries in groups split as graphs of groups, showing how vertex group boundaries influence the overall Morse boundary under certain conditions.

Contribution

It establishes conditions under which Morse boundaries of vertex groups embed into the Morse boundary of the entire group, extending understanding of Morse boundary topology in graph of groups.

Findings

Connected components of Morse boundaries originate from vertex groups.

Under certain conditions, Morse boundaries of vertex groups are topologically embedded.

Edge groups' properties influence the Morse boundary structure.

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 $\partial_MG$.