Linear norm

Jump to: navigation, search

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

$\|xy\| \leq \|x\| + \|y\| \quad (1)$

and the linear growth condition

$\|x^n \| = |n| \|x\| \quad (2)$

for all $x,y \in G$ and $n \in {\bf Z}$.

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

Key lemmas

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

From (2) we of course have

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

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

$n \|x\| = \|x^n \| \leq \|z\| + n \|y\| + \|z^{-1} \|$

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