Linear norm

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This is the wiki page for understanding seminorms of linear growth on a group [math]G[/math] (such as the free group on two generators). These are functions [math]\| \|: G \to [0,+\infty)[/math] that obey the triangle inequality

[math]\|xy\| \leq \|x\| + \|y\| \quad (1)[/math]

and the linear growth condition

[math] \|x^n \| = |n| \|x\| \quad (2) [/math]

for all [math]x,y \in G[/math] and [math]n \in {\bf Z}[/math].

We use the usual group theory notations [math]x^y := yxy^{-1}[/math] and [math][x,y] := xyx^{-1}y^{-1}[/math].


Key lemmas

Henceforth we assume we have a seminorm [math]\| \|[/math] of linear growth. The letters [math]x,y,z,w[/math] are always understood to be in [math]G[/math], and [math]i,j,n,m[/math] are always understood to be integers.

From (2) we of course have

[math] \|x^{-1} \| = \| x\| \quad (3)[/math]
Lemma 1 If [math]x[/math] is conjugate to [math]y[/math], then [math]\|x\| = \|y\|[/math].

Proof: By hypothesis, [math]x = zyz^{-1}[/math] for some [math]z[/math], thus [math]x^n = z y^n z^{-1}[/math], hence by the triangle inequality

[math] n \|x\| = \|x^n \| \leq \|z\| + n \|y\| + \|z^{-1} \|[/math]

for any [math]n \geq 1[/math]. Dividing by [math]n[/math] and taking limits we conclude that [math]\|x\| \leq \|y\|[/math]. Similarly [math]\|y\| \leq \|x\|[/math], giving the claim. [math]\Box[/math]