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Mixed identities, hereditarily separated actions and oscillation

Aleksander Ivanov, Roland Zarzycki

TL;DR

The paper develops a general framework for solvability of mixed identities with constants in groups acting on topological spaces by introducing explicitly oscillating words and the notion of hereditarily separating actions. It shows that if a word has an explicitly oscillating conjugate in a setting with hereditary separation, then the inequality $w(\bar{y})\neq 1$ has a solution, and extends this to systems of inequalities with support-control. It further refines the theory via the Transition procedure to handle non-explicit oscillations and proves density-type results for oscillating words in Polish and locally compact group actions. The work connects oscillation to MIF properties, provides partial MIF results, and applies the framework to Thompson's group $F$ and branch groups, highlighting broad implications for dynamics and model-theoretic aspects of groups acting on spaces.

Abstract

Given a topological $G$-space we consider equations with parameters over $G$. In particular we formulate some very general conditions on words with parameters $w(\bar{y},\bar{g})$ over $G$ which guarantee that the inequality $w(\bar{y},\bar{g})\neq 1$ has a solution in $G$. These results are illustrated in some typical situations, in particular standard actions of Thompson's group $F$ and branch groups are considered. The major results of this paper appeared in some form in Section 2 of the PhD thesis of the second author (avalable at arXiv:1308.6330).

Mixed identities, hereditarily separated actions and oscillation

TL;DR

The paper develops a general framework for solvability of mixed identities with constants in groups acting on topological spaces by introducing explicitly oscillating words and the notion of hereditarily separating actions. It shows that if a word has an explicitly oscillating conjugate in a setting with hereditary separation, then the inequality has a solution, and extends this to systems of inequalities with support-control. It further refines the theory via the Transition procedure to handle non-explicit oscillations and proves density-type results for oscillating words in Polish and locally compact group actions. The work connects oscillation to MIF properties, provides partial MIF results, and applies the framework to Thompson's group and branch groups, highlighting broad implications for dynamics and model-theoretic aspects of groups acting on spaces.

Abstract

Given a topological -space we consider equations with parameters over . In particular we formulate some very general conditions on words with parameters over which guarantee that the inequality has a solution in . These results are illustrated in some typical situations, in particular standard actions of Thompson's group and branch groups are considered. The major results of this paper appeared in some form in Section 2 of the PhD thesis of the second author (avalable at arXiv:1308.6330).
Paper Structure (12 sections, 14 theorems, 102 equations)

This paper contains 12 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Explicitly oscillating words and hereditarily separating actions
  3. Explicitly oscillating words
  4. Hereditarily separating actions by homeomorphisms
  5. Partial MIF
  6. Inequalities in groups with hereditarily separating action
  7. Oscillating words
  8. Solving a system of inequalities in the case of oscillating words
  9. Actions of topological groups
  10. Polish $G$-spaces
  11. Topological actions of locally compact groups
  12. Topological actions of automorphism groups

Key Result

Theorem 2.13

Let $G$ be a group acting on a perfect Hausdorff space $\mathcal{X}$ by homeomorphisms. Let $w(\bar{y})$ be a nontrivial word over $G$ on $t$ variables, $y_{1},\ldots , y_{t}$, which is reduced in $\mathbb{F}_{t}\ast G$ and non-constant (i.e. $w(\bar{y})\notin G$). If $G$ hereditarily separates $\ma

Theorems & Definitions (37)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Example 2.9
  • Example 2.10
  • ...and 27 more