Linear norm: Difference between revisions
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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 {\ | 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 == | ||
| Line 15: | Line 17: | ||
== Key lemmas == | == 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> | |||
Revision as of 12:21, 21 December 2017
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]
