Connected components of Morse boundaries of graphs of groups

Elia Fioravanti and Annette Karrer

Vol. 317 (2022), No. 2, 339–361

DOI: 10.2140/pjm.2022.317.339

Abstract

Let a finitely generated group $G$ split as a graph of groups. If the 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 the Morse boundaries of the vertex groups are topologically embedded in ${\partial }_{M}G$.

Keywords

Morse boundary, graph of groups, totally disconnected, relatively wide

Mathematical Subject Classification

Primary: 20E08, 20F65, 20F67

Milestones

Received: 1 November 2021

Revised: 20 January 2022

Accepted: 22 January 2022

Published: 14 July 2022