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Growth of finitely generated subgroups of the topological full groups of inverse semigroups of bounded type

Zheng Kuang

TL;DR

The paper proves that finitely generated subgroups of the topological full group $\mathsf{F}(G_0)$, when orbit equivalent to an inverse semigroup $G_0$ of bounded type defined by expanding and linearly repetitive tile inflations, have subexponential growth provided the incompressible elements of $G_0$ are finite. It extends prior stationary-Bratteli results to non-stationary tile inflations, linking tile expansion to linear repetitivity and polynomial orbital-graph growth, and then derives a BNZ-style traverse-counting argument to bound the growth by $\gamma_G(R)\preccurlyeq \exp(R^{\alpha})$ with some $\alpha\in(0,1)$. The main theorem is complemented by explicit dihedral-group constructions and fragmentation techniques that produce examples with sublinear to subexponential growth and non-linear orbital graphs, broadening the landscape of intermediate-growth groups acting on Bratteli-path spaces. Together, these results illuminate how geometric expansion, boundary control, and orbit-equivalence interact to constrain growth in topological full groups arising from inverse semigroups of bounded type. The work also connects to continued-fraction phenomena of bounded type through the dihedral-group constructions and demonstrates systematic fragmentation methods to generate a spectrum of intermediate-growth groups in this setting.

Abstract

Given an inverse semigroup $G_0$ of bounded type, we show, along with some other assumptions, that if the set of incompressible elements of $G_0$ is finite, then any finitely generated subgroup $G$ of the topological full group $\mathsf{F}(G_0)$ that is orbit equivalent to $G_0$ has subexponential growth with a bounded power in the exponent.

Growth of finitely generated subgroups of the topological full groups of inverse semigroups of bounded type

TL;DR

The paper proves that finitely generated subgroups of the topological full group , when orbit equivalent to an inverse semigroup of bounded type defined by expanding and linearly repetitive tile inflations, have subexponential growth provided the incompressible elements of are finite. It extends prior stationary-Bratteli results to non-stationary tile inflations, linking tile expansion to linear repetitivity and polynomial orbital-graph growth, and then derives a BNZ-style traverse-counting argument to bound the growth by with some . The main theorem is complemented by explicit dihedral-group constructions and fragmentation techniques that produce examples with sublinear to subexponential growth and non-linear orbital graphs, broadening the landscape of intermediate-growth groups acting on Bratteli-path spaces. Together, these results illuminate how geometric expansion, boundary control, and orbit-equivalence interact to constrain growth in topological full groups arising from inverse semigroups of bounded type. The work also connects to continued-fraction phenomena of bounded type through the dihedral-group constructions and demonstrates systematic fragmentation methods to generate a spectrum of intermediate-growth groups in this setting.

Abstract

Given an inverse semigroup of bounded type, we show, along with some other assumptions, that if the set of incompressible elements of is finite, then any finitely generated subgroup of the topological full group that is orbit equivalent to has subexponential growth with a bounded power in the exponent.
Paper Structure (18 sections, 22 theorems, 64 equations, 11 figures, 2 tables)

This paper contains 18 sections, 22 theorems, 64 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tile inflations and automata
  4. Bounded type
  5. Incompressible elements and traverses
  6. Expanding tile inflations
  7. Exponential growth of diameters
  8. Linearly repetitive actions
  9. Polynomial growth of orbital graphs
  10. Proof of the theorem
  11. Dihedral groups $D_{\infty}$
  12. The construction
  13. Actions $\Omega(\mathsf{B})\curvearrowleft D_{\infty}$ in terms of tile inflations
  14. A fragmentation of $D_{\infty}$ and its subexponential growth estimate
  15. A family of groups with non-linear orbital graphs
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathsf{B}$ be a simple Bratteli diagram. Suppose the inverse semigroup $G_0=\langle \mathcal{S}\rangle$ is defined by an expanding and linearly repetitive tile inflation process and acts minimally on $\Omega(\mathsf{B})$. Let $G<\mathsf{F}(G_0)$ be a finitely generated subgroup that is orbit e

Figures (11)

  • Figure 1: Tower $\mathcal{F}_{0,1}$.
  • Figure 2: Tower $\mathcal{F}_{1,1}$.
  • Figure 4: Finite diagram for $\mathcal{A}_0\prec\mathcal{A}_1$ when $r=5$.
  • Figure 5: Bratteli Diagram $\mathsf{B}_w$
  • Figure 6: Tiles when $w_1=3$ and $w_2=1$.
  • ...and 6 more figures

Theorems & Definitions (72)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Remark 2.2
  • Definition 2.6
  • Definition 2.7
  • ...and 62 more