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Subgroups arising from connected components in the Morse boundary

Annette Karrer, Babak Miraftab, Stefanie Zbinden

Abstract

We study connected components of the Morse boundary and their stabilisers. We introduce the notion of point-convergence and show that if the set of non-singleton connected components of the Morse boundary of a finitely generated group $G$ is point-convergent, then every non-singleton connected component is the (relative) Morse boundary of its stabiliser. The above property only depends on the topology of the Morse boundary and hence is invariant under quasi-isometry. This shows that the topology of the Morse boundary not only carries algebraic information but can be used to detect certain subgroups which in some sense are invariant under quasi-isometry.

Subgroups arising from connected components in the Morse boundary

Abstract

We study connected components of the Morse boundary and their stabilisers. We introduce the notion of point-convergence and show that if the set of non-singleton connected components of the Morse boundary of a finitely generated group is point-convergent, then every non-singleton connected component is the (relative) Morse boundary of its stabiliser. The above property only depends on the topology of the Morse boundary and hence is invariant under quasi-isometry. This shows that the topology of the Morse boundary not only carries algebraic information but can be used to detect certain subgroups which in some sense are invariant under quasi-isometry.
Paper Structure (13 sections, 22 theorems, 10 equations, 5 figures)

This paper contains 13 sections, 22 theorems, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Morseness
  4. The Morse boundary
  5. Morseness and groups
  6. Relative Morse boundary
  7. Point-convergence and its characterisations
  8. Null-families and point-convergence
  9. The anti-core and the core of a non-singleton component
  10. Stabilisers of connected components with finite anti-core
  11. Graph of groups whose family of non-singleton components is point-convergent
  12. Setup and Notation
  13. Proof of \ref{['cor:graph_of_groups']}

Key Result

Theorem 1.1

Let $G$ be a finitely generated group. Let $C$ be a non-singleton connected component of the Morse boundary $\partial_* G$. If $G\cdot C$ is point-convergent, then $(\partial_* H, G) = C$, where $H\leq G$ is the stabiliser of $C$ under the action $G \curvearrowright \partial_* G$.

Figures (5)

  • Figure 1: Definition of $\lambda_{gv}$.
  • Figure 2: A hyperbolic space whose set of boundary components is not point-convergent.
  • Figure 3: Proof of Lemma \ref{['lemma:full_conv']}
  • Figure 4: Proof of \ref{['lemma:close_to_Core']}
  • Figure 5: Proof of \ref{['lemma:close_to_C']}

Theorems & Definitions (55)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7
  • Definition 2.1: Morse gauge
  • Definition 2.2: Morse
  • Remark 2.3
  • Lemma 2.4: CH17
  • Lemma 2.5: Lemma 4.1 of CD16
  • ...and 45 more