Last time, we saw how we could consider Euclidean geometry to be a transition between hyperbolic and spherical geometry. Now we ask, what happens to spaces that are not simply connected under these transitions?

Cylinders

From last time, we consider the family of transformations

\[A_n = \left(\begin{array}{cc} 1 & -n \\ 0 & 1 \end{array} \right) \left(\begin{array}{cc} e^t & 0 \\ 0 & e^t \end{array} \right) \left(\begin{array}{cc} 1 & n \\ 0 & 1 \end{array} \right).\]

As \(n \to \infty\), we have \(A_n \to P = \bigl(\begin{smallmatrix} 1 & s \\ 0 & 1 \end{smallmatrix}\bigr)\).

Consider the transformation \(a_s = \bigl(\begin{smallmatrix}e^s & \\ & e^-s \end{smallmatrix}\bigr)\). What is the space \(H^2 / \langle a_s \rangle\)? The transformation can be viewed in the upper half space model as vertical translation along the y axis, with fundamental domain a semicircular band. The quotient space is thus a hyperbolic cylinder. Note that as \(s\to 0\), we have \(a_s\) approaching the identity.

Now we can conjugate \(a_s\) by the matrix \(\bigl(\begin{smallmatrix}1 & \frac1s \\ 0 & 1\end{smallmatrix}\bigr)\) to get the transformation \(a_s'= \bigl(\begin{smallmatrix}e^s & O(s^2) \\ & e^-s \end{smallmatrix}\bigr)\), a translation about another vertical geodesic. As \(s_0\), the axis of translation moves towards infinity and \(a_s' \to \bigl(\begin{smallmatrix}1 & 2 \\ 0 & 1\end{smallmatrix}\bigr) = a_0'\). So in the limit our hyperbolic transformations (which fix a point on the boundary of the half plane and the point at infinity) approach a parabolic transformation. Our quotient space then becomes a cylinder with a cusp.

Now we consider another family of transformations whose quotients are cylinders. Let

\[M_t = \left(\begin{array}{ccc} \cosh t & 0 & \sinh t \\ 0 & 1 & 0 \\ \sinh t & 0 & \cosh t \end{array}\right) \in \text{PSO}(2,1).\]

The quotient space for \(M_t\) is a cylinder Now if we let \(t \to 0\) the cylinder collapses. So we re-scale by letting \(c_t\) be the matrix with diagonal entries \(\sqrt{t}, \sqrt{t}, \frac 1 t\). Now when we conjugate,

\[c_t^{-1}M_t c_t = \left(\begin{array}{ccc} \cosh t & 0 & \frac{1}{t} \sinh t\\ 0 & 1 & 0\\ t\sinh t & 0 & \cosh t \end{array}\right).\]

Now as \(t\to 0\) we get

\[\left(\begin{array}{ccc} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right).\]

Now with the re-scaling, as we expand the space it looks more and more Euclidean, and in the limit the quotient space is a Euclidean cylinder.

Tori

Now we want to construct hyperbolic and spherical tori that transition to Euclidean tori.

First, a definition. Let \(\Sigma\) be a compact surface, \(X\) a finite collection of points in \(\Sigma\). Then a hyperbolic (or Euclidean, or spherical) cone surface is a hyperbolic structure on \(\Sigma_X = \Sigma \setminus X\) such that for each point \(p\in X\), the completion of the structure at \(p\) looks like

\[ds^2 = dr^2 + \sinh^2 r d\theta^2\]

where \(\theta\) is taken modulo the cone angle \(\alpha \in \mathbb{R}\). In other words, we have a hyperbolic structure where at a finite collection of cone points there is less than a \(2\pi\) angle. When \(\alpha\) is a rational multiple of \(\pi\), this is the same thing as an orbifold.

Now let

\[A_t = \left(\begin{array}{ccc} \cosh t & & \sinh t\\ &1&\\ \sinh t & & \cosh t \end{array}\right)\]

and

\[B_t = \left(\begin{array}{ccc} 1 & & \\ & \cosh t & \sinh t\\ & \sinh t & \cosh t \end{array} \right).\]

These transformations are hyperbolic translations in different directions, and \(\mathbb{H}^2/\langle A_t B_t \rangle\) is a cone torus.

We can repeat our re-scaling trick to take the limit of these structures. When we conjugate \(A_t\) and \(B_t\) by the matrix \(c_t\) defined above, then as \(t \to 0\) we again get Euclidean translations and a quotient space that is a square torus. In fact, by tweaking the construction we can have our cone tori limit to any Euclidean torus.

Borromean Rings

We can give the Borromean rings a hyperbolic structure, but it’s hard. Instead we will give a structure similar to a con structure: a Euclidean structure where the Borromean rings form a singular locus similar to a cone point.

First, we tile \(\mathbb{R}^3\) with cubes, and let \(G_B\) be the group generated by the 180 degree rotations about a line segment across each face of the cube. When we take the quotient of \(mathbb{R}^3\) by this group, we find that these rotational axes form the Borromean rings.

Now, with the faces of the cube split into two by the axes of rotation, we can consider our cube to really be a degenerate dodecahedron. We can build a dodecahedron in hyperbolic space that has the correct angles about the edges corresponding to the rotational axes. Now we can glue the faces together via similar rotations to get our same quotient space. Now we have a hyperbolic structure on \(S^3\) which is branched along the Borromean rings.

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