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Studying Stein's Groups as Topological Full Groups

Owen Tanner

Abstract

We include a class of generalisations of Thompson's group $V$ introduced by Melanie Stein into the growing framework of topological full groups. Like $V$, Stein's groups can be described as certain piecewise linear maps with prescribed slopes and nondifferentiable points. Stein's groups include the class of groups where the group of slopes is generated by multiple integers (which we call Stein's integral groups) and when the group of slopes is generated by a single irrational number, such as Cleary's group, (which we call irrational slope Thompson's groups). We give a unifying dynamical proof that whenever the group of slopes is finitely generated, the simple derived subgroup of the associated Stein's groups is finitely generated. We then study the homology of Stein's groups, building on the work of others. This line of inquiry allows us to compute the abelianisations and rational homology concretely for many examples.

Studying Stein's Groups as Topological Full Groups

Abstract

We include a class of generalisations of Thompson's group introduced by Melanie Stein into the growing framework of topological full groups. Like , Stein's groups can be described as certain piecewise linear maps with prescribed slopes and nondifferentiable points. Stein's groups include the class of groups where the group of slopes is generated by multiple integers (which we call Stein's integral groups) and when the group of slopes is generated by a single irrational number, such as Cleary's group, (which we call irrational slope Thompson's groups). We give a unifying dynamical proof that whenever the group of slopes is finitely generated, the simple derived subgroup of the associated Stein's groups is finitely generated. We then study the homology of Stein's groups, building on the work of others. This line of inquiry allows us to compute the abelianisations and rational homology concretely for many examples.
Paper Structure (13 sections, 45 theorems, 117 equations)

This paper contains 13 sections, 45 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids and Partial Actions
  4. Topological Full Groups
  5. Stein's Groups
  6. Construction of Stein's Groups as Topological Full Groups
  7. Finite Generation of The Derived Subgroup
  8. Expansivity of Partial Transformation Groupoids
  9. Finite Generation of Stein's Groups
  10. Homology of Stein's groups
  11. Initial observations
  12. Homology for Irrational Slope Thompson's groups
  13. Homology for Stein's Integral Groups

Key Result

Theorem 2

Let $\Lambda$ be a subgroup of $(\mathbb{R}_+, \cdot)$ and $\Gamma$ be a $\mathbb{Z} \cdot \Lambda$-submodule and $\ell \in \Gamma$. Then, $D(V(\Gamma,\Lambda,\ell))$ is simple. Moreover, the following are equivalent:

Theorems & Definitions (99)

  • Definition 1: Stein's groups
  • Theorem 2: Theorem \ref{['fg when fg by alg']}
  • Theorem 3
  • Corollary 4
  • Corollary 5: Corollary \ref{['classification for low degree']}
  • Corollary 6: Corollary \ref{['acyclicty for stein']}
  • Example 2.1
  • Example 2.2: Partial Action
  • Example 2.3
  • Definition 2.4: Purely Infinite
  • ...and 89 more