Let $L$ be any real lower triangular matrix with positive diagonal entries (a Cholesky matrix). Let $x$ and $b$ be real vectors. Is the group of actions $(L, b)$ on $x$, $$L x + b$$ amenable?
Asked
Active
Viewed 53 times
1
-
Unless I'm missing something, isn't the statement of the Banach-Tarski paradox that the action of the Euclidean group in $3$ dimensions is not amenable? – Jose Avilez Jan 24 '24 at 16:02
-
@JoseAvilez I don't think that all actions of the Euclidean group like rotations can be expressed like this using a Cholesky matrix L – Jannis Jan 24 '24 at 16:46
1 Answers
0
Yes.
Scaling $Lx$: The group of invertible lower triangular matrices under matrix multiplication is a Borel subgroup of the general linear group. This implies that it is solvable. The group of lower triangular matrices with positive diagonal entries is therefore also solvable since subgroups of solvable groups are solvable.
Shifting $x+b$: The group of vectors under addition is solvable since it is abelian.
The group of actions $Lx+b$ on $x$ is the semidirect product of the shifting and scaling groups. The semidirect product of solvable groups is solvable. Every solvable group is amenable.
Jannis
- 163