Difference between revisions of "Linear norm"
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:<math> \| x^{-1} [x,y]^2 \| \leq \| yx^{-1} y^{-1} \| + \|xyx^{-2} y^{-1} x\|</math> | :<math> \| x^{-1} [x,y]^2 \| \leq \| yx^{-1} y^{-1} \| + \|xyx^{-2} y^{-1} x\|</math> | ||
and the claim follows from Lemma 1 and (3). <math>\Box</math> | and the claim follows from Lemma 1 and (3). <math>\Box</math> | ||
+ | |||
+ | '''Corollary 7'''. For any <math>m,k \geq 1</math>, one has | ||
+ | :<math> \| x^m [x,y]^k \| \leq \frac{1}{2} ( \| x^{m-1} [x,y]^k \| + \|x^{m+1} [x,y]^{k-1} \| )</math>. | ||
+ | |||
+ | '''Proof'''. ??? | ||
== Iterations == | == Iterations == |
Revision as of 12:57, 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].
Contents
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]\| \|[/math] of linear growth. The letters [math]s,t,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]
An equivalent form of the lemma is that
- [math] \|xy\| = \|yx\| \quad (4).[/math]
We can generalise Lemma 1:
Lemma 2. If [math]x^i[/math] is conjugate to [math]wy[/math] and [math]x^j[/math] is conjugate to [math]zw^{-1}[/math], then [math] \|x\| \leq \frac{1}{|i+j|} ( \|w\| + \|z\| )[/math].
Proof. By hypothesis, [math]x^i = s wy s^{-1}[/math] and [math]x^j = t zw^{-1} t^{-1}[/math] for some [math]s,t[/math]. For any natural number [math]n[/math], we then have
- [math] x^{in} x^{jn} = s wy \dots wy s^{-1} t zw^{-1} \dots zw^{-1} t^{-1}[/math]
where the terms [math]wy, zw[/math] are each repeated [math]n[/math] times. By Lemma 1, conjugation by [math]w[/math] does not change the norm. From many applications of this and the triangle inequality, we conclude that
- [math] |i+j| n \|x\| = \| x^{in} x^{jn} \| \leq \|s\| + n \|y\| + \|s^{-1} t\| + n \|z\| + \|t^{-1}\|.[/math]
Dividing by [math]n[/math] and sending [math]n \to \infty[/math], we obtain the claim. [math]\Box[/math]
Corollaries
Corollary 0. The eight commutators [math][x^{\pm 1}, y^{\pm 1}], [y^{\pm 1}, x^{\pm 1}][/math] all have the same norm.
Proof. Each of these commutators is conjugate to either [math][x,y][/math] or its inverse. [math]\Box[/math]
Corollary 1. The function [math]n \mapsto \|x^n y\|[/math] is convex in [math]n[/math].
Proof. [math]x^n y[/math] is conjugate to [math]x (x^{n-1} y)[/math] and to [math](x^{n+1} y) x^{-1}[/math], hence by Lemma 2
- [math]\| x^n y \| \leq \frac{1}{2} (\| x^{n-1} y \| + \| x^{n+1} y \|),[/math]
giving the claim. [math]\Box[/math]
Corollary 2. For any [math]k \geq 1[/math], one has
- [math]\| [x,y] \| \leq \frac{1}{2k+2} (\| [x^{-1},y^{-1}]^k x^{-1} \| + \| [x,y]^k x \|).[/math]
Thus for instance
- [math]\| [x,y] \| \leq \frac{1}{4} (\| [x^{-1},y^{-1}] x^{-1} \| + \| [x,y] x \|).[/math]
Proof. [math][x,y]^{k+1}[/math] is conjugate both to [math]x(y[x^{-1},y^{-1}]^k x^{-1}y^{-1})[/math] and to [math](y^{-1} [x,y]^k xy)x^{-1}[/math], hence by Lemma 2
- [math] \| [x,y] \| \leq \frac{1}{2k+2} ( \| y[x^{-1},y^{-1}]^k x^{-1} \| + \| (y^{-1} [x,y]^k xy)x^{-1}\|)[/math]
giving the claim by Lemma 1. [math]\Box[/math]
Corollary 3. One has
- [math] \|[x,y]^2 x\| \leq \frac{1}{2} ( \| x y^{-1} [x,y] \| + \| xy [x,y] \| ).[/math]
Proof. [math][x,y]^2 x[/math] is conjugate both to [math]y (x^{-1} y^{-1} [x,y] x^2)[/math] and to [math](x[x,y]xyx^{-1}) y^{-1}[/math], hence by Lemma 2
- [math] \displaystyle \|[x,y]^2 x\| \leq \frac{1}{2} ( \|x^{-1} y^{-1} [x,y] x^2\| + \|x[x,y]xyx^{-1}\|)[/math]
giving the claim by Lemma 1. [math]\Box[/math]
Corollary 4. One has
- [math]\| [x,y] x\| \leq \frac{1}{4} ( \| x^2 y [x,y] \| + \| xy^{-1} x [x,y] \| ).[/math]
Proof. [math]([x,y] x)^2[/math] is conjugate both to [math]y^{-1} (x [x,y] x^2 y x^{-1})[/math] and to [math](x^{-1} y^{-1} x [x,y] x^2) y[/math], hence Lemma 2
- [math]\| [x,y] x\| \leq \frac{1}{4} ( \| x [x,y] x^2 y x^{-1} \| + \| x^{-1} y^{-1} x [x,y] x^2 \| ),[/math]
giving the claim by Lemma 1. [math]\Box[/math]
Corollary 5. One has
- [math] \|[x,y] x\| \leq \|x\| + \frac{1}{2} \| [x^2, y] \|[/math].
Proof. [math][x,y]x[/math] is conjugate to both [math]x [x^{-2},y^{-1}][/math] and to [math](y^{-1} x^2 y) x^{-1}[/math], hence by Lemma 2
- [math]\| [x,y] x\| \leq \frac{1}{2} ( \| [x^{-2}, y^{-1}] \| + \| y^{-1} x^2 y \| ),[/math]
giving the claim by Lemma 1 and Corollary 0. [math]\Box[/math]
Corollary 6. One has
- [math] \| [x,y]\| \leq \frac{1}{4} ( \| x\| + \| [x^2,y] \| + \| [x,y] x\| ) [/math].
Proof. From Lemma 2 we have
- [math] \| [x,y] \| \leq \frac{1}{4} ( \| x^{-1} [x,y]^2 \| + \| [x,y] x \| ).[/math]
Since [math]x^{-1} [x,y]^2[/math] is conjugate to [math](yx^{-1} y^{-1}) (xyx^{-2} y^{-1} x)[/math], we have
- [math] \| x^{-1} [x,y]^2 \| \leq \| yx^{-1} y^{-1} \| + \|xyx^{-2} y^{-1} x\|[/math]
and the claim follows from Lemma 1 and (3). [math]\Box[/math]
Corollary 7. For any [math]m,k \geq 1[/math], one has
- [math] \| x^m [x,y]^k \| \leq \frac{1}{2} ( \| x^{m-1} [x,y]^k \| + \|x^{m+1} [x,y]^{k-1} \| )[/math].
Proof. ???