Linear norm
This is the wiki page for understanding seminorms of linear growth on a group [math]\displaystyle{ G }[/math] (such as the free group on two generators). These are functions [math]\displaystyle{ \| \|: G \to [0,+\infty) }[/math] that obey the triangle inequality
- [math]\displaystyle{ \|xy\| \leq \|x\| + \|y\| \quad (1) }[/math]
and the linear growth condition
- [math]\displaystyle{ \|x^n \| = |n| \|x\| \quad (2) }[/math]
for all [math]\displaystyle{ x,y \in G }[/math] and [math]\displaystyle{ n \in {\bf Z} }[/math].
We use the usual group theory notations [math]\displaystyle{ x^y := yxy^{-1} }[/math] and [math]\displaystyle{ [x,y] := xyx^{-1}y^{-1} }[/math].
Threads
- https://terrytao.wordpress.com/2017/12/16/bi-invariant-metrics-of-linear-growth-on-the-free-group/, Dec 16 2017.
- Bi-invariant metrics of linear growth on the free group, II, Dec 19 2017.
Key lemmas
Henceforth we assume we have a seminorm [math]\displaystyle{ \| \| }[/math] of linear growth. The letters [math]\displaystyle{ x,y,z,w }[/math] are always understood to be in [math]\displaystyle{ G }[/math], and [math]\displaystyle{ i,j,n,m }[/math] are always understood to be integers.
From (2) we of course have
- [math]\displaystyle{ \|x^{-1} \| = \| x\| \quad (3) }[/math]
- Lemma 1 If [math]\displaystyle{ x }[/math] is conjugate to [math]\displaystyle{ y }[/math], then [math]\displaystyle{ \|x\| = \|y\| }[/math].
Proof: By hypothesis, [math]\displaystyle{ x = zyz^{-1} }[/math] for some [math]\displaystyle{ z }[/math], thus [math]\displaystyle{ x^n = z y^n z^{-1} }[/math], hence by the triangle inequality
- [math]\displaystyle{ n \|x\| = \|x^n \| \leq \|z\| + n \|y\| + \|z^{-1} \| }[/math]
for any [math]\displaystyle{ n \geq 1 }[/math]. Dividing by [math]\displaystyle{ n }[/math] and taking limits we conclude that [math]\displaystyle{ \|x\| \leq \|y\| }[/math]. Similarly [math]\displaystyle{ \|y\| \leq \|x\| }[/math], giving the claim. [math]\displaystyle{ \Box }[/math]
