---
tag: DB1BZZJN86
kind: statement
role: lemma
name: Whitehead's "Peak Reduction Lemma"
sources:
  - wikipedia: en:60327286@r1358691180
    locator: Whitehead's algorithm, section Whitehead's algorithm > Whitehead's "Peak Reduction Lemma"
    note: adopted
    mention: wikipedia://en/60327286#NVJP
    sections: [Statement]
inventory: [W97H9M892W]
---

## Statement

The following statement is referred to as Whitehead's "Peak Reduction Lemma", see Proposition 4.20 in  and Proposition 1.2 in:

Let $w\in F_n$. Then the following hold:

1. If $[w]$ is not automorphically minimal, then there exists a Whitehead automorphism $\tau\in \operatorname{Aut}(F_n)$ such that $\|\tau(w)\|_X< \|w\|_X$.
1. Suppose that $[w]$ is automorphically minimal, and that another conjugacy class $[w']$ is also automorphically minimal. Then $\operatorname{Aut}(F_n)w=\operatorname{Aut}(F_n)w'$ if and only if $\|w\|_X=\|w'\|_X$ and there exists a finite sequence of Whitehead moves $\tau_1,\dots, \tau_k\in\operatorname{Aut}(F_n)$ such that $$\tau_k\cdots \tau_1(w)=w'$$ and $$\|\tau_i\cdots\tau_1(w)\|_X=\|w\|_X \text{ for } i=1,\dots, k.$$

Part (1) of the Peak Reduction Lemma implies that a conjugacy class $[w]$ is Whitehead minimal if and only if it is automorphically minimal.