The real affine group is amenable and nonunimodular

The multiplication is associative because both bracketings of $(a,b),(a^{\mathrm{\prime }},b^{\mathrm{\prime }}),(a^{\mathrm{\prime }\mathrm{\prime }},b^{\mathrm{\prime }\mathrm{\prime }})$ give $(a{a}^{\mathrm{\prime }}{a}^{\mathrm{\prime }\mathrm{\prime }},a{a}^{\mathrm{\prime }}{b}^{\mathrm{\prime }\mathrm{\prime }}+a{b}^{\mathrm{\prime }}+b)$; $(1,0)$ is the identity and $(a,b{)}^{-1}=(a^{-1},-b\mathrm{/}a)$. Each coordinate of multiplication is a sum or product of continuous coordinate maps, and each inverse coordinate is a quotient with denominator $a>0$, so the group operations are continuous by [F1]. The set $G$ is open in ${\mathbb{R}}^2$. At every point of this open set, the compact-closure neighbourhood base in the locally compact Hausdorff Euclidean space supplies a compact neighbourhood still contained in $G$; Hausdorffness is inherited. Thus $G$ is a locally compact Hausdorff topological group.

The coordinate formulas give the subgroup laws and commutativity of $N$ and $H$. The conjugation formula in [F3] shows $gN{g}^{-1}=N$ for every $g=(a,b)$, so $N$ is normal, since $t\mapsto at$ is onto $\mathbb{R}$ for $a>0$. Also $(1,b)(a,0)=(a,b)$ for every $(a,b)\in G$, so every group element is a product from $NH$.

For each $n\in N$, the maps $x\mapsto nx$ and ${\mathrm{id}}_X$ are continuous into the Hausdorff space $X$, so their agreement set is closed by [F4]. Their intersection is $X^N$, hence $X^N$ is closed; it is compact because $X$ is compact, and it is convex because each action map is affine.

By [A1] and [A2], the positive density $\omega =a^{-2}\mathrm{\mid }dadb\mathrm{\mid }$ on the open smooth manifold $G$ defines a compact-finite Radon Borel measure $\mu (E):={\int }_E{a}^{-2}d{\lambda }_2(a,b)$ for Borel $E\subseteq G$. This measure is nonzero: on $K=[1,2]\times [0,1]\subset G$ the density is at least $1\mathrm{/}4$ and ${\lambda }_2(K)=1$, so [F13] gives $\mu (K)\ge 1\mathrm{/}4$.

The action restricted to the abelian topological group $N$ is continuous and affine on the nonempty compact convex set $X$. By [F5], it has a fixed point, which belongs to $X^N$. Therefore $X^N$ is nonempty.

The set $X^N$ is invariant under $H$: if $x\in X^N$, $h\in H$, and $n\in N$, then $n(hx)=h((h^{-1}nh)x)=hx$, since $h^{-1}nh\in N$ by normality. The restricted $H$-action on $X^N$ is continuous and affine, because it is the restriction of the given continuous affine action.

Fix $g_0=(a_0,b_0)\in G$. Left translation $T_{g_0}(a,b)=(a_{0}a,a_{0}b+b_0)$ is a $C^1$ diffeomorphism of $G$ onto itself with Jacobian determinant $a_0^2$. For every Borel $E\subseteq G$, [F9] gives $\mu (g_{0}E)={\int }_E(a_{0}a{)}^{-2}{a}_0^{2}d{\lambda }_2(a,b)=\mu (E)$. Thus $\mu$ is a left Haar measure by [F10].

The abelian topological group $H$ acts continuously and affinely on the nonempty compact convex set $X^N$ by step 2.2. By [F5], there is $x\in X^N$ fixed by every element of $H$.

For $g_0=(a_0,b_0)$, right multiplication by $g_0^{-1}$ is the $C^1$ diffeomorphism $T_{g_0^{-1}}(a,b)=(a\mathrm{/}{a}_0,b-a{b}_0\mathrm{/}{a}_0)$, whose inverse is $(A,B)\mapsto (A{a}_0,B+A{b}_0)$ and has Jacobian determinant $a_0$. Applying [F9] to the nonnegative positive and negative parts of the real and imaginary parts of $f\in C_c(G)$ gives ${\int }_{G}f(x{g}_0^{-1})d\mu (x)={\int }_{G}f(A,B)(A{a}_0{)}^{-2}{a}_{0}d{\lambda }_2(A,B)=a_0^{-1}{\int }_{G}f(A,B)A^{-2}d{\lambda }_2(A,B)$. By [F11] and the left Haar conclusion of step 2.3, ${\mathrm{\Delta }}_G(g_0)=a_0^{-1}$.

This $x$ is fixed by both $N$ and $H$. Since $G=NH$ by step 1.2, it is fixed by every element of $G$. The action was arbitrary, so $G$ has the fixed point property; [F6] makes $G$ amenable in the sense of [F7].

Taking $g_0=(2,0)$ in step 3.2 gives ${\mathrm{\Delta }}_G(g_0)=1\mathrm{/}2\mathrm{\ne }1$; therefore $G$ is nonunimodular by [F12]. Step 4.1 proves it is amenable, completing both claims in the Statement.