Topological full groups of minimal subshifts and quantifying local embeddings into finite groups

TL;DR

This paper explores the LEF property in topological full groups of minimal subshifts, relating LEF growth to dynamical complexity and constructing groups with novel LEF growth types.

Contribution

It establishes bounds on LEF growth using recurrence and complexity functions, and constructs a continuum of finitely generated LEF groups with distinct growth behaviors.

Findings

LEF growth can be bounded by recurrence and complexity functions

Constructed groups with previously unseen LEF growth types

Demonstrated a continuum of distinguishable finitely generated LEF groups

Abstract

We investigate quantitative aspects of the LEF property for subgroups of the topological full group $[[ \sigma ]]$ of a two-sided minimal subshift over a finite alphabet, measured via the LEF growth function. We show that the LEF growth of $[[ \sigma ]]^{\prime}$ may be bounded from above and below in terms of the recurrence function and the complexity function of the subshift, respectively. As an application, we construct groups of previously unseen LEF growth types, and exhibit a continuum of finitely generated LEF groups which may be distinguished from one another by their LEF growth.