I have a few questions about amenability at infinity for locally compact, second countable, Hausdorff topological groups. Recall that a locally compact group $G$ is said to be amenable at infinity if there exists a topologically amenable action (in the sense of http://www.univ-orleans.fr/mapmo/membres/anantharaman/publications/Exactness02.pdf) of $G$ on some compact Hausdorff space $X$.
Permanence property:
Q 1. What kinds of permanence properties do we know about amenability at infinity for locally compact, second countable non-discrete groups? e.g. If $\Gamma$ is a Lattice in $G$, which is amenable at infinity, does it imply that $G$ itself is amenable at infinity?
Examples:
Q 2. Which locally compact non-discrete groups are known to be amenable at infinity? e.g. Are Haagerup groups amenable at infinity?
Q 3. Let $G$ be a locally compact, second countable Hausdorff topological group, which acts amenably on a compact Hausdorff space $X$. Does the transformation groupoid $X\rtimes G$ admit a continuous proper negative type function $\psi$?
negative type means
1) $\psi(x,e)=0$ for all $x\in X$;
2) $\psi(x,g)=\psi(g^{-1}x,g^{-1})$ for all $(x,g)\in X\times G$;
3) $\sum_{i,j=1}^nt_it_j \psi(g_i^{-1}x,g_i^{-1}g_j)\leq 0$ for all $t_1,\ldots,t_n$ in $\mathbb{R}$ satisfying $\sum_{i=1}^n t_i=0$, $g_i\in G$ and $x\in X$.
