Fundamental Groups of Disjointly Tree-Graded Spaces

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.

Figures (6)

• Figure 1: A disjointly tree-graded space where the pieces are illustrated as circles or enlarged points.
• Figure 2: A disjointly tree-graded space $(X,\mathscr{P})$ where $\mathbf{Pc}(X)$ is not uniformly $1$-$UV_0$.
• Figure 3: Open convex open sets $O_1,O_2,O_3,\dots$ enumerating some of the connected components of $\mathbb{D}\backslash K$. The map $g:K\to X$ sends the boundaries $\partial O_n$ into $A$ by null-homotopic loops that shrink in diameter as $n\to\infty$.
• Figure 4: $e_n$ is chosen in $\eta^{-1}(T\backslash V)$ to separate $\rho(h)$ and $\rho\bigl(\operatorname{hull}(S_n)\bigr)$.
• Figure 5: A set $h\in H_{\mathbf{Pc}}$ such that the components $J_1,J_2,J_3$ of $\partial h\cap S^1$ are mapped into the piece $P$ of $X$ non-trivially by $\alpha$. The image of $\partial h$ under quotient map $\rho_1$ is a simple closed curve where the closures of the three components of $\partial h\backslash (J_1\cup J_2\cup J_3)$ are elements of $H_{\partial h}$ and thus identified to points.

Theorems & Definitions (156)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Definition 2.2
• Definition 3.1
• Remark 3.2
• Remark 3.3