Fundamental Groups of Disjointly Tree-Graded Spaces

Tree-graded spaces are a generalization of $\mathbb{R}$-trees and play an important role in describing the large-scale geometry of relatively hyperbolic groups. We consider a subclass of tree-graded spaces that we call “disjointly tree-graded spaces,” determined by maps to $\mathbb{R}$-trees. We characterize the fundamental group of a disjointly tree-graded space $(X,\mathcal{P})$ in terms of the fundamental groups of its pieces. Our results apply even in cases where neither $X$ nor its pieces are locally simply connected. In particular, we show that if the pieces are uniformly $1$-$U{V}_0$, then the fundamental group of a disjointly tree-graded space embeds into the inverse limit of the free products of the fundamental groups of finitely many pieces.