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Generalized Residual Finiteness of Groups

Nic Brody, Kasia Jankiewicz

Abstract

A countable group is residually finite if every nontrivial element can act nontrivially on a finite set. When a group fails to be residually finite, we might want to measure how drastically it fails - it could be that only finitely many conjugacy classes of elements fail to act nontrivially on a finite set, or it could be that the group has no nontrivial actions on finite sets whatsoever. We define a hierarchy of properties, and construct groups which become arbitrarily complicated in this sense.

Generalized Residual Finiteness of Groups

Abstract

A countable group is residually finite if every nontrivial element can act nontrivially on a finite set. When a group fails to be residually finite, we might want to measure how drastically it fails - it could be that only finitely many conjugacy classes of elements fail to act nontrivially on a finite set, or it could be that the group has no nontrivial actions on finite sets whatsoever. We define a hierarchy of properties, and construct groups which become arbitrarily complicated in this sense.
Paper Structure (15 sections, 10 theorems, 7 equations)

This paper contains 15 sections, 10 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Background on ordinals and cardinals
  3. Ordinals
  4. Ordinal arithmetic
  5. Cardinals
  6. $(\alpha,\kappa)$-residual finiteness
  7. Motivation
  8. Definition
  9. Properties
  10. Actions on rooted trees
  11. Faithful actions on rooted finite valence trees
  12. Rooted $\alpha$-trees
  13. Actions on rooted $\alpha$-trees
  14. A construction of $\omega\cdot n$-residually finite groups
  15. Application to profinite rigidity

Key Result

Theorem 1.1

For every $n\in \mathbb Z$, where $n\geq 1$, there exists a finitely generated group $G_n$ which is $\omega\cdot n$-residually finite, but not $\omega\cdot(n-1)$-residually finite.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Example 3.1: Higman
  • Example 3.2: Deligne78, see also morris2009lattice
  • Example 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Example 3.7
  • ...and 16 more