Difference between revisions of "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
 
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\|</math>
+
:<math>\|xy\| \leq \|x\| + \|y\| \quad (1)</math>
  
 
and the linear growth condition
 
and the linear growth condition
  
:<math> \|x^n \| = |n| \|x\| </math>
+
:<math> \|x^n \| = |n| \|x\| \quad (2) </math>
  
for all <math>x,y \in G</math> and <math>n \in {\bb Z}</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>.
 +
 
 
== Threads ==
 
== Threads ==
  
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== Key lemmas ==
 
== Key lemmas ==
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 +
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>
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 +
:'''Lemma 1''' If <math>x</math> is conjugate to <math>y</math>, then <math>\|x\| = \|y\|</math>.
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 +
'''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
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 +
:<math> n \|x\| = \|x^n \| \leq \|z\| + n \|y\| + \|z^{-1} \|</math>
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 +
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>

Revision as of 12:21, 21 December 2017

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].

Threads

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]