The "Seed" $\mathcal S = (\mathcal I, \Gamma, \Pi)$

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Consider a triple $\mathcal S = (\mathcal I, \Gamma, \Pi)$, where $\mathcal I$ is a compactified 4-sheeted branched cover of $\Bbb C$, $\Gamma$ is its branching locus, and $\Pi$ is a finite nonabelian piecewise monodromy group.

Specifically, let $\mathcal{O}_1, \dots, \mathcal{O}_4$ be four spindle orbifolds (topologically, Riemann spheres $\widehat{\mathbb{C}}$ each with two cone points). Let their union form the 2-complex:

$$\mathcal{I} = \bigcup_{j=1}^{4} \mathcal{O}_j$$

These four surfaces intersect precisely along a 1-dimensional locus $\Gamma$, which serves as the 1-skeleton of the complex. Combinatorially, $\Gamma$ is a 4-regular octahedral graph ($|V| = 6$, $|E| = 12$). Geometrically, $\mathcal{I}$ is intrinsically parameterized within a bounding 3-dimensional cube, with the $6$ vertices (0-cells) of $\Gamma$ sinking into the interior, located exactly on the local coordinate planes $x, y, z = 1/2$. Let $P = \{p_{j,1}, p_{j,2}\}_{j=1}^4$ be the set of the $8$ cone points across the four orbifolds. By construction, $P \cap \Gamma = \emptyset$.

We prescribe a particular set of markings for the cells. Each of the $\mathcal O_j$ gets a distinct color. Each coordinate cycle of $\Gamma$ gets a distinct color. To see the latter we decompose $\Gamma$ into $\Gamma = \Gamma_x \cup \Gamma_y \cup \Gamma_z$. The total partition gives $3+4=7$ distinct colors. By inspection, $\Pi \cong G_{2\times 2}$ where $G_{2\times 2}$ is the Rubik's pocket cube group. This is a diagram of the 'seed' $\mathcal S$:



To get this piecewise monodromy, we define $\Pi = \langle \pi_x, \pi_y, \pi_z \rangle$, where the generators represent quarter-turn piecewise isometries along the coordinate planes $x, y, z = 1/2$. The generators possess order 4, satisfying $\pi_x^4 = \pi_y^4 = \pi_z^4 = \mathrm{id}$. And $\Pi$ possesses a semi-direct product structure $\mathbb{Z}_3^7 \rtimes S_8$, yielding the rigid, finite nonabelian group that was proposed at the start.

The motivation for constructing this rigid framework is to study the spectrum of a transform, defined by the kernel $\varphi_s(x) = e^{s/\ln x}$, as it operates across the ramified sheets of $\mathcal{I}$. 

Let $e$ be one of the 12 edges of $\Gamma$. We parameterize this edge with a local coordinate $x \in (0, 1]$, where $x \to 0$ approaches one of the 6 vertices (the cone point singularity where the spindle orbifolds intersect) and $x=1$ is the boundary of the local fundamental domain.

A function on $\Gamma$ must act like a $\Pi$-automorphic form. Its local behavior near the vertex is dictated by the spectral parameters of the space. Take a simple base function representing a single spectral component of the automorphic form near the singularity:

$$f(x) = x^{a-1}$$

where $a > 0$ is a spectral eigenvalue parameter dictated by the invariant subspace of the pocket cube group $\Pi$.

Now apply a transform with the specific kernel $\varphi_s(x) = e^{s/\ln x}$ by integrating over the edge from $0$ to $1$:

$$\mathcal{Z}\{f\}(s) = \int_{0}^{1} x^{a-1} e^{s/\ln x} \, dx$$

To evaluate this, we make a change of variables to pull the function out of the log domain. Let $u = -\ln x$. This gives us $x = e^{-u}$ and the differential $dx = -e^{-u} \, du$. For the bounds: as $x \to 0$, $u \to \infty$, and when $x = 1$, $u = 0$. Substituting these into the integral, we get:

$$\mathcal{Z}\{f\}(s) = \int_{\infty}^{0} (e^{-u})^{a-1} e^{-s/u} (-e^{-u}) \, du$$

The negative sign flips the bounds of integration, and we combine the exponential terms:

$$\mathcal{Z}\{f\}(s) = \int_{0}^{\infty} e^{-au} e^{-s/u} \, du$$

$$\mathcal{Z}\{f\}(s) = \int_{0}^{\infty} e^{-\left( au + \frac{s}{u} \right)} \, du$$ 

This integral is the integral representation for the modified Bessel function of the second kind. Evaluating it yields:

$$\mathcal{Z}\{f\}(s) = 2 \sqrt{\frac{s}{a}} K_1(2\sqrt{as})$$

This is the radial profile of an eigenfunction required by J. Cheeger's spectral analysis near a cone point. 

Moreover we know that $\varphi_s(x)$ satisfies the diffusion equation:

$$ s \frac{\partial^2}{\partial s^2}\varphi_s(x) = - x \frac{\partial}{\partial x}\varphi_s(x) $$

which suggests that the transform $\mathcal Z$ is actively flowing the spectral info of the $\Pi$-automorphic form, located on $\Gamma$, to the cone points where it manifests as the radial eigenfunction contribution:

$$\mathcal{Z}\{f\}(s) = 2 \sqrt{\frac{s}{a}} K_1(2\sqrt{as})$$

This form is interesting because it satisfies a (dispersive) partial differential equation. We let $\mathcal{Z}\{f\}(s):=F_s(a)$. Then:

$$ s^2 \frac{\partial^3}{\partial s^3}F_s(a) = a^2 \frac{\partial}{\partial a} F_s(a) $$

 And this can be interpreted as a propagating wave outward from a cone point, which decays sufficiently to remain $L^2$ integrable.

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