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I am trying to see whether the pair $(GL(2,\mathbb{Q})^+,SL(2,\mathbb{Z}))$ is amenable in the following sense:

Let $H$ be a closed subgroup of a locally compact group $G$. The pair $(G,H)$ is called amenable if it satisfies the fixed point property; that is if $G$ acts on a compact convex subset $Q$ of a locally convex vector space as a group of affine transformations and there is a fixed point for the restriction of this action to $H$, then there is a fixed point for the action of $G$ as well.

We know that $GL(2,\mathbb{Q})^+$ (considered as a discrete group) can be written as the semidirect product $SL(2,\mathbb{Q})\rtimes \mathbb{Q}^+$. So if we prove that the pair $(SL(2,\mathbb{Q}),SL(2,\mathbb{Z}))$ is amenable then the amenability of the former pair follows. On the other hand, we know that the pair $(SL(2,\mathbb{R}),SL(2,\mathbb{Z}))$ is amenable (it seems it is due to Kazhdan). But I don't know how to conclude the amenability of $(SL(2,\mathbb{Q}),SL(2,\mathbb{Z}))$ from this, because $SL(2,\mathbb{Q})$ is not a closed subgroup of $SL(2,\mathbb{R}))$.

Can anyone help me to prove or disprove the amenability of the pair $(SL(2,\mathbb{Q}),SL(2,\mathbb{Z}))$? or at least give me a hint or a helpful reference?

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  • $\begingroup$ This is usually called "$H$ is co-amenable in $G$". $\endgroup$ – YCor Sep 24 '13 at 7:12
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    $\begingroup$ The answer is negative: indeed otherwise by considering the dense homomorphism into $PGL_2(\mathbb{Q}_p)$, we would infer that $PGL_2(\mathbb{Z}_p)$ is co-amenable in the latter. Since it is compact, this would mean that $PGL_2(\mathbb{Q}_p)$ is amenable, a contradiction. $\endgroup$ – YCor Sep 24 '13 at 7:14
  • $\begingroup$ @YvesCornulier: I think the continuity of co-amenability is an immediate corollary of the fixed point property. So ignore my previous comment. BTW, I appreciate your help. $\endgroup$ – user23860 Sep 24 '13 at 7:48

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