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Amenability problem for Thompson's group $F$: state of the art

Victor Guba

Abstract

This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions.

Amenability problem for Thompson's group $F$: state of the art

Abstract

This is a survey of our recent results on the amenability problem for Thompson's group . They mostly concern esimating the density of finite subgraphs in Cayley graphs of for various systems of generators, and also equations in the group ring of . We also discuss possible approaches to solve the problem in both directions.
Paper Structure (12 sections, 34 theorems, 33 equations, 2 figures)

This paper contains 12 sections, 34 theorems, 33 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Cayley graphs
  3. Automata, density, and isoperimetric constants
  4. Amenability of semigroups and groups
  5. Rooted binary trees and forests
  6. Thompson's group $F$
  7. Density of Cayley graphs
  8. Equations in group rings and their systems
  9. Miscellaneous
  10. Structure of Cayley graphs
  11. Algorithm for the word problem
  12. Perspectives

Key Result

Proposition 1.1

A group $G$ with finite set of generators $A$ is amenable if and only if its Cheeger isoperimetric constant iz zero: $\iota_*(G;A)=0$.

Figures (2)

  • Figure 1: The vertices $a,b,c$ in the left Cayley graph of $F$ in standard generators.
  • Figure 2: Fragment of the left Cayley graph of $F$ in generators $\{x_0,x_1,x_2\}$ for any special forest.

Theorems & Definitions (34)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 2.1: Gu21a
  • Theorem 2.2
  • Proposition 2.3: Gu22
  • Proposition 2.4: Gu22
  • Theorem 2.5: Gu22
  • Theorem 2.6: Gu22
  • Theorem 2.7: Gu23
  • ...and 24 more