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The "spread" of Thompson's group $F$

Gili Golan

Abstract

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element $g\in G$ has a co-generator in $G$ (i.e., an element which together with $g$ generates $G$). Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. In 2022, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. Since the abelianization of Thompson's group $F$ is $\mathbb{Z}$, it cannot be $\frac{3}{2}$-generated. However, we recently proved that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part of a generating pair of $\mathbb{Z}^2$ is part of a generating pair of $F$. A natural generalization of $\frac{3}{2}$-generation is the notion of spread. Recall that the spread of a group $G$ is the supremum over all integers $k$ such that every $k$ non-trivial elements of $G$ have a common co-generator in $G$. The uniform spread of a group $G$ is the supremum over all integers $k$ for which there exists a conjugacy class $C\subseteq G$ such that every $k$ non-trivial elements of $G$ have a common co-generator which belongs to $C$. In this paper we study modified versions of these notions for Thompson's group $F$.

The "spread" of Thompson's group $F$

Abstract

Recall that a group is said to be -generated if every non-trivial element has a co-generator in (i.e., an element which together with generates ). Thompson's group was proved to be -generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic -generated group. In 2022, Bleak, Harper and Skipper proved that Thompson's group is also -generated. Since the abelianization of Thompson's group is , it cannot be -generated. However, we recently proved that Thompson's group is "almost" -generated in the sense that every element of whose image in the abelianization forms part of a generating pair of is part of a generating pair of . A natural generalization of -generation is the notion of spread. Recall that the spread of a group is the supremum over all integers such that every non-trivial elements of have a common co-generator in . The uniform spread of a group is the supremum over all integers for which there exists a conjugacy class such that every non-trivial elements of have a common co-generator which belongs to . In this paper we study modified versions of these notions for Thompson's group .
Paper Structure (9 sections, 21 theorems, 10 equations, 1 figure)

This paper contains 9 sections, 21 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. F as a group of homeomorphisms
  4. Elements of F as pairs of binary trees
  5. Choosing elements of $F$
  6. Generating Thompson's group $F$
  7. Subgroups $H$ of $F$ whose closure contains $[F,F]$
  8. Preliminary lemmas
  9. Proof of the main result

Key Result

Lemma 2

Let $S$ be a finite subset of $F$ and let $g\in F$ be such that $\pi(g)$ is a common co-generator of the elements of $\pi(S)$ in $\mathbb{Z}^2$. Assume that the following conditions hold. Then the elements of $S$ do not have a common co-generator in $F$ that is a conjugate of $g$.

Figures (1)

  • Figure 1: (A) The reduced tree-diagram of $x_0$. (B) The reduced tree-diagram of $x_1$. In both figures, $T_+$ is on the left and $T_-$ is on the right.

Theorems & Definitions (44)

  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3
  • Corollary 4
  • Remark 5
  • proof
  • Corollary 6
  • Remark 7
  • Corollary 8
  • ...and 34 more