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Almost finiteness and groups of dynamical origin

Petr Naryshkin, Spyridon Petrakos

TL;DR

Addressing the problem of when actions of amenable groups on compact spaces are almost finite, the authors introduce the notion of good subgroups and show that amenable actions with such subgroups have dynamical comparison; for free actions on finite-dimensional spaces, this yields almost finiteness. The paper demonstrates that many groups of dynamical origin, including topological full groups of Cantor minimal systems and the Basilica group, satisfy the hypotheses, thereby obtaining classifiable crossed products in the finite-dimensional setting. A key insight is that micro-supported actions automatically generate good subgroups, expanding applicability to weakly branch groups and related Thompson-like, IET, and iterated monodromy groups. These results provide a concrete route to verifying the Elliott classification program for a wide family of amenable crossed products.

Abstract

We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces, and show that it implies comparison when the acting group is amenable. As a consequence, free actions on finite-dimensional spaces of many notable amenable groups of dynamical origin are almost finite. For instance, this applies to topological full groups of Cantor minimal systems and the Basilica group. In particular, minimal such actions give rise to classifiable crossed products.

Almost finiteness and groups of dynamical origin

TL;DR

Addressing the problem of when actions of amenable groups on compact spaces are almost finite, the authors introduce the notion of good subgroups and show that amenable actions with such subgroups have dynamical comparison; for free actions on finite-dimensional spaces, this yields almost finiteness. The paper demonstrates that many groups of dynamical origin, including topological full groups of Cantor minimal systems and the Basilica group, satisfy the hypotheses, thereby obtaining classifiable crossed products in the finite-dimensional setting. A key insight is that micro-supported actions automatically generate good subgroups, expanding applicability to weakly branch groups and related Thompson-like, IET, and iterated monodromy groups. These results provide a concrete route to verifying the Elliott classification program for a wide family of amenable crossed products.

Abstract

We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces, and show that it implies comparison when the acting group is amenable. As a consequence, free actions on finite-dimensional spaces of many notable amenable groups of dynamical origin are almost finite. For instance, this applies to topological full groups of Cantor minimal systems and the Basilica group. In particular, minimal such actions give rise to classifiable crossed products.
Paper Structure (5 sections, 7 theorems, 25 equations)

This paper contains 5 sections, 7 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Amenability, almost finiteness, and comparison
  4. Micro-supported actions
  5. Comparison from good subgroups and the main results

Key Result

Theorem A

Suppose that $\Gamma$ is an amenable group that admits a faithful micro-supported (Definition def:microsup) action on a Hausdorff Baire space with no isolated pointsthis assumption can be slightly weakened, see Remark rem: microsup weakening. Then any free action of $\Gamma$ on a finite-dimensional

Theorems & Definitions (24)

  • Conjecture A
  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 14 more