non-amenable subgroup of an amenable locally compact groups

Question

I begin by recall this two know facts:

1- Every subgroup of a discrete amenable group is amenable

2-Every closed subgroup of a locally compact amenable group is amenable.

I need an example of an locally compact amenable group $G$ with a non-amenable subgroup $H$.

More precisely, i need an example to confirm that the hypothesis closed is crucial for the fact 2.

Thank for any help.

Answer 1

The free group on two generators $\mathbb{F}_2$, which is non-amenable, is a subgroup of $SO_3$ the group of rotations of $\mathbb{R}^3$, which is compact and thus amenable.

An interesting historical fact, I think that this was the example (used in the Banach-Tarski paradox) that motivated von Neumann to defined amenability and nonamenability.