A A trace-coordinate check of the WP factor

A separate two-dimensional calculation is useful for checking the factor $2$ in the local densities. Let $A,B,C,D\in\mathrm{SL}_2(\mathbb R)$ satisfy $ABCD=I$, and set $$a=\mathop{\mathrm{tr}}A,\ b=\mathop{\mathrm{tr}}B,\ c=\mathop{\mathrm{tr}}C,\ d=\mathop{\mathrm{tr}}D, \qquad x=\mathop{\mathrm{tr}}(AB),\ y=\mathop{\mathrm{tr}}(AC),\ z=\mathop{\mathrm{tr}}(BC).$$ With the four peripheral traces fixed, the character surface satisfies the Fricke relation $$\begin{aligned} \Phi(x,y,z)={}&x^2+y^2+z^2+xyz-(ab+cd)x-(ac+bd)y\\ &-(ad+bc)z+a^2+b^2+c^2+d^2+abcd-4=0. \end{aligned}$$ This is the standard four-holed-sphere trace surface; see [59].

On a chart where $x=2\cosh(\ell/2)>2$, put $s=\sinh(\ell/2)$ and $X=AB$. Twisting across the $AB$ cuff fixes $A,B$ and conjugates $C,D$ by $$\exp\left(\frac{\tau}{2}H_X\right), \qquad H_X=\frac{X-(x/2)I}{s}.$$ The eigenvalues of $H_X$ are $1,-1$, so this is a displacement of length $\tau$ along the cuff axis. Differentiating $y=\mathop{\mathrm{tr}}(AC)$ gives $$\frac{\partial y}{\partial\tau} =\frac1{2s}\mathop{\mathrm{tr}}\bigl(A[X,C]\bigr) =-\frac{\Phi_z}{2s}.$$ For the last equality, Cayley–Hamilton gives $\mathop{\mathrm{tr}}(A^2BC)=ad-z$, while the two-matrix trace identity gives $\mathop{\mathrm{tr}}(ACAB)=xy-bc+z$. Their difference is $ad+bc-xy-2z=-\Phi_z$.

Since $\mathop{}\!\mathrm dx=s\,\mathop{}\!\mathrm d\ell$ and $\omega_{\mathrm{WP}}=\mathop{}\!\mathrm d\ell\wedge\mathop{}\!\mathrm d\tau$, $$\boxed{\mathop{}\!\mathrm dV_{\mathrm{WP}}=\frac{2|\mathop{}\!\mathrm dx\wedge\mathop{}\!\mathrm dy|}{|\Phi_z|}.}$$(A.2) Equivalently, on this twist chart, $\mathop{}\!\mathrm dx\wedge\mathop{}\!\mathrm dy=-\Phi_z\,\mathop{}\!\mathrm d\ell\wedge\mathop{}\!\mathrm d\tau/2$. Reversing the chosen twist direction reverses the signed form but leaves the displayed positive density unchanged. The hyperbolic application requires the smooth geometric component of the relative character surface and $\Phi_z\ne0$ for $(x,y)$ to be coordinates. If $\Phi_z=0$ but $\nabla\Phi\ne0$, another pair can be chosen; if all three partial derivatives vanish, the character surface is singular and this inversion argument does not supply a chart. In particular, arbitrary real solutions of $\Phi=0$ are not automatically holonomies of four-holed spheres. The formula is local on the marked character surface and does not describe a global mapping-class fundamental domain.