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Bounded displacement permutations on tree-like spaces

Samuel M. Corson

TL;DR

The paper addresses the structure of the bounded displacement permutation group on tree-like spaces by proving that, under finiteness and coarse-tree hypotheses, infinitely divisible elements are torsion. The authors introduce a general framework using a quasi-isometric embedding into a bounded-valence tree and a separation property to force torsion, then apply it to deduce the main result. This yields concrete corollaries for virtually free groups and extends prior work by Suchkov, Shlepkin, and Taysnyov, including recovering the no-copy-of-$\mathbb{Q}$ result for $\mathfrak{Bd}(\mathbb{Z})$. The findings deepen understanding of the subgroup structure of $\mathfrak{Bd}(X,d)$ in spaces that resemble trees at large scale and have controlled local growth, with potential implications for quasi-isometric rigidity and related group-theoretic properties.

Abstract

It is shown that if a metric space exhibits certain finiteness and tree-like properties, then elements of its group of bounded displacement which are infinitely divisible are also torsion. This extends a result of N. M. Suchkov, A. A. Shlepkin, and D. A. Taysnyov.

Bounded displacement permutations on tree-like spaces

TL;DR

The paper addresses the structure of the bounded displacement permutation group on tree-like spaces by proving that, under finiteness and coarse-tree hypotheses, infinitely divisible elements are torsion. The authors introduce a general framework using a quasi-isometric embedding into a bounded-valence tree and a separation property to force torsion, then apply it to deduce the main result. This yields concrete corollaries for virtually free groups and extends prior work by Suchkov, Shlepkin, and Taysnyov, including recovering the no-copy-of- result for . The findings deepen understanding of the subgroup structure of in spaces that resemble trees at large scale and have controlled local growth, with potential implications for quasi-isometric rigidity and related group-theoretic properties.

Abstract

It is shown that if a metric space exhibits certain finiteness and tree-like properties, then elements of its group of bounded displacement which are infinitely divisible are also torsion. This extends a result of N. M. Suchkov, A. A. Shlepkin, and D. A. Taysnyov.
Paper Structure (2 sections, 3 theorems, 4 equations)

This paper contains 2 sections, 3 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Proof

Key Result

Theorem 1

Suppose that for metric space $(X, d)$, simplicial tree $T$, and $f:X \rightarrow T$ there exist $\ell, m, q, \in \mathbb{Z}^+$ such that Then each infinitely divisible element in $\mathfrak{B}\mathfrak{d}(X, d)$ is torsion.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 2
  • Proposition 3
  • proof
  • proof : Proof of Theorem \ref{['maintheorem']}