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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 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?


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 $x\in X$.

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What is a connection between this property and G. Yu's property A? – Mark Sapir Mar 23 '13 at 2:24
Mark: I think it is the same property. – Misha Mar 23 '13 at 3:46
Yu's property A is only defined for discrete metric space. Higson proved that they are the same for discrete groups, but I only interest in non-doscrete cases. see… – m07kl Mar 23 '13 at 7:17
@m07kl: Did you ask Erik Guentner? What did he say? – Mark Sapir Mar 23 '13 at 14:53
@Mark Sapir: No, Shall I send him an email or maybe he is on Mathoverflow? – m07kl Mar 23 '13 at 16:32

Q1: Answer can be founded in

Q2: I think it is still open.

Q3: Answer is Yes. It is proved by Tu see LEMME 3.5 in "La conjecture de Baum-Connes pour les feuilletages moyennables. K-Theory 17 (1999), no. 3, 215--264"

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