A Note on the Metric of Thompson's group V

TL;DR

The paper addresses refining the metric bound for Thompson's group $V$ by incorporating permutation structure via cluster analysis. It defines the cluster count $B(x)$ for an element $x\in V$ and proves a bound $\frac{N(x)}{C}\le ||x|| \le C\left(N(x)+B(x)\log B(x)\right)$, improving Birget's previous estimate and matching the known behavior for $F$ and $T$. The main contribution is a method that collapses clusters using left and right multiplications by elements of $F$, yielding a diagram with $B(x)$ carets and allowing the Birget bound to apply, thereby giving better estimates for many elements of $V$. However, the bound is not sharp in general, as shown by a counterexample, indicating room for further refinement toward a permutation-informed metric $D(x)$. The results advance sharp metric estimates within Thompson's groups and guide future work on distortion and permutation-aware distance quantities.

Abstract

In this short note, a bound on the word metric for Thompson's group V given by Birget in 2004 is improved to a new bound, which agrees with the known bounds for Thompson's groups F and T.

Figures (3)

• Figure 1: An example of an element of $V$ with seven leaves (six carets) but only four clusters. The clusters are marked with dotted lines on the diagram, and observe that they coincide with the connected components of the graph of the corresponding map.
• Figure 2: The process of collapsing the clusters of the element in Figure \ref{['element']}. In the last diagram each leaf has been labelled with all the labels of the cluster collapsing into it.
• Figure 3: The counterexample to show the bound is not sharp. The unlabelled leaves are not permuted.

Theorems & Definitions (2)

• Definition 1.1
• Theorem 1.2