Conjugator length in Thompson's groups

TL;DR

This paper establishes that in Thompson's groups, the conjugator length function grows quadratically, providing both upper and lower bounds, which advances understanding of their algebraic complexity.

Contribution

The paper proves that Thompson's groups have quadratic conjugator length functions, offering new bounds and insights into their conjugacy problem complexity.

Findings

Conjugator length in Thompson's groups is quadratic in the length of elements.

Existence of conjugate pairs with quadratic minimal conjugator length.

Results apply to groups F, T, and V.

Abstract

We prove Thompson's group $F$ has quadratic conjugator length function. That is, for any two conjugate elements of $F$ of length $n$ or less, there exists an element of $F$ of length $O(n^2)$ that conjugates one to the other. Moreover, there exist conjugate pairs of elements of $F$ of length at most $n$ such that the shortest conjugator between them has length $\Omega(n^2)$. This latter statement holds for $T$ and $V$ as well.