Is there an example of a non compact, semisimple, amenable Lie group?

Question

By semisimple I mean the real Lie algebra of $G$ is semisimple. I guess there is not but I can't formulate a rigorous argument.

Answer 1

Let $G$ be a noncompact semisimple Lie group. If $G$ is connected, then it contains an isomorphic copy of $\mathbb{F}_2$, the free group with 2 generators (see Theorem 3.9 in the book by Patterson), thus not amenable.

If $G$ is not connected, then let $G_e$ be the connected component of identity. $G_e$ is an open subgroup, so the factor group $G/G_e$ is discrete. $G$ is amenable iff both $G_e$ and $G/G_e$ are amenable. By definition, $G$ and $G_e$ has the same solvable radical (i.e., the largest solvable connected closed normal subgroup), so $G_e$ is also semisimple. Thus, the connected semisimple Lie group $G_e$ is amenable iff $G_e$ is compact. In summary, if $G$ is not connected, then $G$ is amenable iff $G_e$ is compact and $G/G_e$ is an amenable discrete group.

To find an example of a non-connected non-compact semisimple Lie group, take a compact semisimple Lie group $H$ and an infinite amenable discrete group $D$, and define the cartesian product $G=H\times D$.