Juschenko–Matte Bon–Monod–de la Salle, Corollary 1.4, case of a free affine action — an extensively amenable action carrying a free affine cocycle action makes the group amenable

Let a group $G$ act on a set $X$, let $L$ be an abelian group, and let $c : G \to L^{(X)}$ be a map into the finitely supported functions $X \to L$ (X →₀ L) such that

• every $c(g)$ is supported in a set $Y \subseteq X$;
• $c(gh) = c(g) + g_* c(h)$ for all $g, h \in G$, where $g_*\varphi$ is the push-forward of $\varphi$ along $x \mapsto g \cdot x$ (Finsupp.mapDomain (g • ·));
• the affine action $\varphi \mapsto c(g) + g_*\varphi$ of $G$ on $L^{(X)}$ is free: $c(g) + g_*\varphi = \varphi$ only when $g = 1$.

If the action of $G$ on $X$ is extensively amenable relative to $Y$ (IsExtensivelyAmenableOn G X Y), then $G$ is amenable (Garrido.IsAmenable G).

Juschenko, Matte Bon, Monod and de la Salle, p. 3: “Corollary 1.4. Let $G \curvearrowright X$ be an extensively amenable action and let $F : \mathbf I \to \mathbf{Amen}$ be any functor. A subgroup $H$ of $F(X) \rtimes G$ is amenable as soon as the intersection $H \cap (\{1\} \times G)$ is so.” and “Remark 1.5. A particular case in which this criterion applies is when one is able to construct a twisted embedding $G \hookrightarrow F(X) \ltimes G$ of the form $g \mapsto (c_g, g)$ with the property that $\{g \in G : c_g = 1\}$ is an amenable subgroup of $G$. We then say that $c : G \to F(X)$, $g \mapsto c_g$ is a $F(X)$-cocycle with amenable kernel. The conclusion is then that $G$ is amenable.”

Formalization note. This is the case of Remark 1.5 for the functor $X \mapsto L^{(X)}$, under a stronger hypothesis: every point stabilizer of the affine action is trivial, not only the kernel $\{g : c(g) = 0\}$ amenable. Extensive amenability is taken relative to a set $Y$ containing the supports of the cocycle, as in the bundle's IsExtensivelyAmenableOn.