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$E\mathcal{Z}$-boundaries, splittings over finite subgroups, and dense amalgams

Mateusz Kandybo, Jacek Świątkowski

Abstract

The dense amalgam is an operation (introduced in arXiv:1410.4989) which to any finite collection of metrizable compacta associates canonically some new highly disconnected compact metrisable space in which embedded copies of the initial spaces are appropriately uniformly and disjointly distributed. We show that in the very general framework of $E\mathcal{Z}$-boundaries (unifying many frameworks such as Gromov boundaries, CAT(0)-boundaries, systolic boundaries, etc.), any boundary $Z$ of an infinitely ended group $Γ$ equipped with an appropriate splitting along finite subgroups has a form of the dense amalgam of the limit sets in $Z$ of the factor subgroups of this splitting.

$E\mathcal{Z}$-boundaries, splittings over finite subgroups, and dense amalgams

Abstract

The dense amalgam is an operation (introduced in arXiv:1410.4989) which to any finite collection of metrizable compacta associates canonically some new highly disconnected compact metrisable space in which embedded copies of the initial spaces are appropriately uniformly and disjointly distributed. We show that in the very general framework of -boundaries (unifying many frameworks such as Gromov boundaries, CAT(0)-boundaries, systolic boundaries, etc.), any boundary of an infinitely ended group equipped with an appropriate splitting along finite subgroups has a form of the dense amalgam of the limit sets in of the factor subgroups of this splitting.
Paper Structure (17 sections, 27 theorems, 42 equations)

This paper contains 17 sections, 27 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs of groups
  4. EZ-boundaries and limit sets
  5. Dense amalgams
  6. An outline of the proof of Theorem A
  7. Separation
  8. R-separation
  9. Separation in graphs of groups
  10. Separation by compact subsets in EZ-structures
  11. Separation in EZ-structures for graphs of groups with finite edge groups
  12. Points in Z
  13. Classification of points in Z
  14. Some topological properties of Z
  15. Finite vertex groups
  16. ...and 2 more sections

Key Result

Lemma 2.1 .8

Let $(\mathcal{G},Y)$ be a non-elementary graph of groups with all of the edge groups being finite. Let $\Gamma\coloneqq\pi_1(\mathcal{G}, Y, T)$, and let $\widetilde{X} \coloneqq\widetilde{X}(\mathcal{G}, Y, T)$ be the Bass-Serre tree.

Theorems & Definitions (70)

  • Definition 2.1 .1: Paths and branches
  • Definition 2.1 .2: Graph of groups
  • Definition 2.1 .3: Fundamental group of a graph of groups
  • Definition 2.1 .4: Bass-Serre tree
  • Remark 2.1 .5
  • Definition 2.1 .6: Elementary collapse
  • Definition 2.1 .7: Non-elementary graph of groups
  • Lemma 2.1 .8
  • Definition 2.2 .9
  • Remark 2.2 .10
  • ...and 60 more