Theorem 6.4 of Juschenko–Matte Bon–Monod–de la Salle on the breakpoints, for H(ℤ) — a subgroup of H(ℤ) is amenable if and only if its action on P_ℤ is extensively amenable

Let $K$ be a subgroup of the homeomorphisms of the projective line $\mathbf P^1 = \mathbb R \cup \{\infty\}$ contained in Monod's group $H(\mathbb Z)$ (Monod.H ⊥, where ⊥ is the smallest subring of $\mathbb R$, namely $\mathbb Z$). Then $K$ is amenable (Garrido.IsAmenable K) if and only if its action on $\mathbf P^1$ by evaluation is extensively amenable relative to $P_{\mathbb Z}$ (IsExtensivelyAmenableOn K (OnePoint ℝ) (Monod.P ⊥)), the set of fixed points in $\mathbf P^1$ of hyperbolic elements of $\mathrm{SL}_2(\mathbb Z)$, where the elements of $H(\mathbb Z)$ may break.

For $K = H(\mathbb Z)$ this restates Monod's Problem 12 (ThompsonAmenability.not_isAmenable_H_bot): $H(\mathbb Z)$ is non-amenable if and only if its action on $P_{\mathbb Z}$ is not extensively amenable.

Juschenko, Matte Bon, Monod and de la Salle, p. 23: “Theorem 6.4. A subgroup $H_1$ of $H$ is amenable if and only if $H_1 \curvearrowright \mathbf R$ is extensively amenable.” For the subgroups of $H(\mathbb Z)$ this statement asks for extensive amenability on $P_{\mathbb Z}$ in place of $\mathbb R$, and it implies the form on $\mathbb R$: ThompsonAmenability.isAmenable_iff_isExtensivelyAmenableOn_of_le_H_bot.

Route. The amenable direction holds for every action of an amenable group. For the other, let $c(g)(x) = \log g'_+(x) - \log g'_-(x)$, the jump of the logarithmic derivative of $g$ at $x$, supported in the breakpoints of $g$, hence in $P_{\mathbb Z}$; at $\infty$ record instead the translation by which $g$ acts near $\infty$. Then $c(gh) = c(g) + g_* c(h)$, and the affine action $\varphi \mapsto c(g) + g_*\varphi$ on the finitely supported real functions is free: an element fixing some $\varphi$ is the identity near $\infty$ and has zero jump at each of its fixed points, while at the right end $b$ of its support it would pass from a non-trivial element of $\mathrm{PSL}_2(\mathbb Z)$ fixing the irrational $b$, which is hyperbolic, to the identity, a non-zero jump. ThompsonAmenability.isAmenable_of_isExtensivelyAmenableOn_of_cocycle then gives amenability.