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Sofic approximations and quantitative measure couplings

Amandine Escalier

Abstract

Measure equivalence was introduced by Gromov as a measured analogue of quasi-isometry. Unlike the latter, measure equivalence does not preserve the large scale geometry of groups and happens to be very flexible in the amenable world. Indeed the Ornstein-Weiss theorem shows that all infinite countable amenable groups are measure equivalent to the group of integers. To refine this equivalence relation and make it responsive to geometry, Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of measure equivalence. They also defined a relaxed version of this notion called quantitative measure subgroup coupling. In this article we offer to answer the inverse problem of the quantification (find a group admitting a measure subgroup coupling with a prescribed group with prescribed quantification) in the case of the lamplighter group.

Sofic approximations and quantitative measure couplings

Abstract

Measure equivalence was introduced by Gromov as a measured analogue of quasi-isometry. Unlike the latter, measure equivalence does not preserve the large scale geometry of groups and happens to be very flexible in the amenable world. Indeed the Ornstein-Weiss theorem shows that all infinite countable amenable groups are measure equivalent to the group of integers. To refine this equivalence relation and make it responsive to geometry, Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of measure equivalence. They also defined a relaxed version of this notion called quantitative measure subgroup coupling. In this article we offer to answer the inverse problem of the quantification (find a group admitting a measure subgroup coupling with a prescribed group with prescribed quantification) in the case of the lamplighter group.
Paper Structure (35 sections, 12 theorems, 36 equations, 7 figures, 1 table)

This paper contains 35 sections, 12 theorems, 36 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantitative measure couplings
  3. Quantitative measure subgroup couplings
  4. Monotonicity of the isoperimetric profile
  5. Background on the inverse problem
  6. Main result
  7. Structure of the paper and strategy of the proof
  8. Background on diagonal products
  9. Definition of diagonal products
  10. General definition
  11. Relative commutators subgroups
  12. The expanders case
  13. Range of an element
  14. From the isoperimetric profile to the group
  15. Definition of the diagonal product
  16. ...and 20 more sections

Key Result

Theorem 1.3

Let $m\geq 2$ be an integer and let $L_m:=(\mathbb{Z}/m\mathbb{Z}) \wr \mathbb{Z}$. For all $\rho\in \mathcal{C}$ there exists a group $G$ such that

Figures (7)

  • Figure 1: Representation of $(\boldsymbol{g},t)=((a_m\delta_0)_m,0) ((b_m\delta_{k_m})_m,0)(0,3)$ when $k_m=2^m$.
  • Figure 2: Representation of $(\boldsymbol{g},0)$ defined in \ref{['Ex:Commutateur']}
  • Figure 3: An element of $\Delta$
  • Figure 4: Data needed to characterized $\boldsymbol{g}$ such that $\mathrm{range}(\boldsymbol{g})\subset [0,6]$ when $k_m=2^m$
  • Figure 5: Definition of $g\cdot x$
  • ...and 2 more figures

Theorems & Definitions (43)

  • Theorem 1.3
  • Definition 2.1: BZ
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Claim 2.7: BZ
  • Example 2.8
  • Proposition 2.9
  • ...and 33 more