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Parabolic cut pairs in boundaries of relatively hyperbolic groups

Kushlam Srivastava

TL;DR

The paper addresses inseparable parabolic cut pairs on Bowditch boundaries of relatively hyperbolic groups. It develops a robust combination theorem that, from a graph of relatively hyperbolic vertex groups with a fixed signature, yields a new relatively hyperbolic group whose boundary contains inseparable parabolic cut pairs, and proves that all such phenomena arise from this construction. It then provides two explicit topological realizations of the boundary in terms of vertex-group boundaries: a completion of a tree-system of cut pairs and an associated inverse limit, unifying the boundary description with a tree-of-spaces framework. Finally, a decomposition theorem shows that any relatively hyperbolic group boundary with inseparable parabolic cut pairs can be obtained from the combination construction, establishing a complete correspondence between boundary topology and the underlying group action on a tree.

Abstract

Parabolic cut pairs in the boundaries of relatively hyperbolic group are a new and previously unexplored phenomenon. In this paper, we give a way to create examples of relatively hyperbolic groups with parabolic cut pairs on their boundary via a combination theorem, which states that a group $G$, splitting as a graph of relatively hyperbolic groups with certain conditions, is relatively hyperbolic with inseparable parabolic cut pairs on the boundary $\partial(G,\mathcal{P})$. We also prove that all relatively hyperbolic groups with inseparable parabolic cut pairs in their boundaries arise via this combination theorem. Świątkowski gives a topological description of combining boundaries of vertex groups. Unfortunately, his method cannot be applied for fundamental reasons in this setting. We instead give two explicit topological descriptions of the boundary in terms of boundaries of the vertex groups.

Parabolic cut pairs in boundaries of relatively hyperbolic groups

TL;DR

The paper addresses inseparable parabolic cut pairs on Bowditch boundaries of relatively hyperbolic groups. It develops a robust combination theorem that, from a graph of relatively hyperbolic vertex groups with a fixed signature, yields a new relatively hyperbolic group whose boundary contains inseparable parabolic cut pairs, and proves that all such phenomena arise from this construction. It then provides two explicit topological realizations of the boundary in terms of vertex-group boundaries: a completion of a tree-system of cut pairs and an associated inverse limit, unifying the boundary description with a tree-of-spaces framework. Finally, a decomposition theorem shows that any relatively hyperbolic group boundary with inseparable parabolic cut pairs can be obtained from the combination construction, establishing a complete correspondence between boundary topology and the underlying group action on a tree.

Abstract

Parabolic cut pairs in the boundaries of relatively hyperbolic group are a new and previously unexplored phenomenon. In this paper, we give a way to create examples of relatively hyperbolic groups with parabolic cut pairs on their boundary via a combination theorem, which states that a group , splitting as a graph of relatively hyperbolic groups with certain conditions, is relatively hyperbolic with inseparable parabolic cut pairs on the boundary . We also prove that all relatively hyperbolic groups with inseparable parabolic cut pairs in their boundaries arise via this combination theorem. Świątkowski gives a topological description of combining boundaries of vertex groups. Unfortunately, his method cannot be applied for fundamental reasons in this setting. We instead give two explicit topological descriptions of the boundary in terms of boundaries of the vertex groups.

Paper Structure

This paper contains 12 sections, 54 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgments
  4. Preliminaries
  5. Relatively hyperbolic groups
  6. Bowditch boundaries and Peano continua
  7. Trees of metrizable spaces
  8. Combining fine hyperbolic graphs
  9. Joining metric compacta along cut pairs as a completion
  10. Joining metric compacta along cut pairs as an inverse limit
  11. Combining Bowditch boundaries along cut pairs
  12. Decomposing along parabolic cut pairs

Key Result

Lemma 2.3

Let $K$ be a $(G,\mathcal{P})$ graph for some relatively hyperbolic pair $(G,\mathcal{P})$. If $e=(x,y)\notin \mathop{\mathrm{Edge}}\nolimits(K)$ add the edges $ge=(gx,gy)$ for all $g\in G$ to construct a new graph $K\cup Ge$. Then $K\cup Ge$ is also a $(G,\mathcal{P})$ graph.

Figures (3)

  • Figure 1: The graph $K$ is constructed by adding pipe edges between a neighboring reservoir and junction. Then collapsing the pipe edges, we get $\overline{K}$.
  • Figure 2:
  • Figure 3: Consider $F',F\in \mathcal{F}$ such that $F'$ contains two vertices from $V$ and $F$ contains one vertex from $V$. The figure describes the kettlebell spaces $M_{F'}^*$ and $M_F^*,$ along with the bonding map $f_{F'F}$ which collapses the dark gray vertex space in $M_{F'}^*$ to the thick peripheral arc in $M_F^*$.

Theorems & Definitions (125)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • ...and 115 more