Fundamental Groups of Disjointly Tree-Graded Spaces

TL;DR

This paper investigates the fundamental groups of disjointly tree-graded spaces, extending understanding of their algebraic structure and embedding properties, especially when pieces are not locally simply connected.

Contribution

It characterizes the fundamental group of disjointly tree-graded spaces in terms of their pieces, even without local simple connectivity, and describes embedding into inverse limits.

Findings

Fundamental group embeds into inverse limits of free products of piece groups.

Results apply to spaces where neither the space nor pieces are locally simply connected.

Provides a new framework for understanding large-scale geometry of relatively hyperbolic groups.

Abstract

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 $(X,\mathscr{P})$ 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 $1$-$UV_0$, 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.