# Linear norm

From Polymath Wiki

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]

and the linear growth condition

- [math] \|x^n \| = |n| \|x\| [/math]

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