###Geometric Transitions

Let’s start by considering the behavior of triangles in the hyperbolic plane. While large triangles differ greatly from being Euclidean (they are thin), the smaller the triangle, the more approximately Euclidean it looks. The same kind of story plays out on the sphere: large triangles “bow outwards” but small triangles are approximately Euclidean.

In fact, if we were to take a triangle in the hyperbolic plane and shrink it, it would appear to be converging onto a Euclidean triangle, but would be shrinking in size so fast that in the limit it would no longer be a triangle, but a point. However, if we looked at our sequence of shrinking triangles through higher and higher powered microscopes (so they appeared to remain approximately the same size) we would observe our hyperbolic triangles get progressively thicker and limit onto a Euclidean one.

However, if we actually want to carry out the above scenario, we can’t achieve our magnification by a simple rescaling (as there are no similarities in the hyperbolic plane other than isometries), so we will in fact need to construct and utilize different copies of the hyperbolic plane within projective 2-space, one for each triangle to live in.

The idea that our above observations are hinting at is that Euclidean geometry reached by the limit of a family of different hyperbolic spaces (and likewise for spherical geometry). In fact, we will see that Euclidean geometry is a transitional geometry lying between hyperbolic and spherical. But to do this, we first need some more precise definitions.

A $$(G,X)$$-structure is a pair consisting of a Lie group $$G$$, a smooth manifold $$X$$ equipped with a transitive analytic $$G$$-action. A geometry $$(H,Y)$$ is said to be a subgeometry of $$(G,X)$$ if $$H\subset G$$ is a closed subgroup and $$Y\subset X$$ is an open submanifold.

For example, hyperbolic geometry can be realized itself as $$(\textrm{PSO}(n,1),\mathbb{H}^n)$$ where here $$\mathbb{H}^n$$ is a unit $$n$$-ball in $$\mathbb{R}P^n$$ and the action of $$\textrm{PSO}(n,1)$$ is by projective transformations. This setup as stated makes it clear that hyperbolic geometry is a subgeometry of projective geometry $$(\textrm{PGL}(n+1,\mathbb{R}),\mathbb{R}P^n)$$. Likewise, Euclidean geometry $$(\textrm{Isom}(\mathbb{E}^n),\mathbb{E}^n)$$ and elliptic geometry $$(\textrm{PO}(n+1),\mathbb{R}P^n)$$ are both subgeometries of Projective Geometry.

So, clearly projective geometry is a reasonable spot to try and make sense of the claim that Euclidean geometry is a transitional geometry between hyperbolic and spherical (ok, well really elliptic…). But to do so we still need to make sense of what it means for a sequence of subgeometries to converge to a geometry. Naively we would expect this to mean something like “$$(G_n,X_n)\to (G,X)$$ in $$(\tilde{G},\tilde{X})$$ if $$G_n\to G$$ and $$X_n\to X$$,” and make sense of this we need to understand how closed subgroupups $$G_n\subset\tilde{G}$$ and subsets $$X_n\subset\tilde{X}$$ converge.

The important case is that of the groups, and here we will mean convergence in the Chabuty topology, which is a topology one can put on the closed subgroups of a Lie group making it into a compact space. The (kind of technical) definition of this topology is given by writing down a basis as follows. Let $$G$$ be a Lie group, and $$C(G)$$ the set of closed subgroups of $$G$$. Let $$H$$ be a closed subgroup, $$K\subset G$$ be compact, and $$U$$ an open neighborhood of the identity in $$G$$. Then the basic open sets for our topology can be described as $$N_{K,U}(H)=\{D\in C(G): D\cap K\subset HU,\; H\cap K\subset DU\}$$. Given this topology, we have $$H_i\to H$$ iff for all $$x\in H$$ there is a sequence $$x_i\in H_i$$ such that $$x_i\to x$$, and if $$x_i\in H_i$$ is a sequence which converges to $$x\in G$$ then $$x\in H$$.

For example, if we let $$H_n$$ be the subgroup of $$\mathbb{R}^2$$ generated by $$e_1$$ and $$ne_2$$, $$H_n\to \langle e_1\rangle$$. However things aren’t always this simple: if $$K_n=\langle e_1, \frac{1}{n}e_2\rangle$$ then in fact (maybe surprisingly at first ) $$K_n\to \mathbb{Z}e_1\oplus \mathbb{R}e_2$$.

Before returning to the idea of a transitional or limit geometry, lets look quickly at two more examples that will be relevant. Let $$A_0=\left\{\left(\begin{array}{cc}e^t& 0\\ 0&e^{-t}\end{array}\right): t\in \mathbb{R}\right \}$$

and let $$A_n$$ be the subgroup of $$\textrm{SL}(2,\mathbb{R})$$ conjugate to $$A_0$$ by $$\left(\begin{array}{cc}1&n\\0&1\end{array}\right)$$. Then if $$A=\left\{\left(\begin{array}{cc}1&s\\0&1\end{array}\right):s\in \mathbb{R}\right\}$$, we have $$A_n\to A$$ in the Chabuty topology. This example shows that a sequence of groups of hyperbolic isometries can converge to a group generated parabolics in the limit.

One further example is given by the following sequence: if $$R_0=\left\{\left(\begin{array}{cc}\cos t &\sin t\\-\sin t &\cos t\end{array}\right):t\in\mathbb{R}\right\}$$ and $$R_n$$ is the congugate of $$R_0$$ by $$\left(\begin{array}{cc}n&0\\0&\frac{1}{n}\end{array}\right)$$. Then $$A_n\to A$$ where $$A$$ is the group $$A=\left\{\left(\begin{array}{cc}1&s\\0&1\end{array}\right):s\in\mathbb{R}\right\}$$. This example shows a sequence of groups of ellptics can also converge to a group of parabolics.

We are now in a good position to define what precisely we mean by a limit of geometric structures: a sequence $$(H_n,Y_n)$$ of subgeometries of $$(G,X)$$ is said to converge to the geometry $$(L,Z)$$ if $$H_n\to L$$ if $$z\in Z$$ then $$z\in Y_n$$ for all sufficiently large $$n$$. Because the two important examples we gave above had to to with sequence of groups that are conjugate to each other, we further will say that $$(L,Z)$$ is a conjugacy limit of $$(H_n, Y_n)$$ if there is some sequence $$g_n\in G$$ such that $$(g_nHg_n^{-1}, g_nY)\to (L,Z)$$ in the sense above.

This lets us make precise our claim that Euclidean geometry is transitional between hyperbolic and elliptic. Namely, we can say that $$(\textrm{Isom}(\mathbb{E}^n, \mathbb{E}^n)$$ is a conjugacy limit of both $$(\textrm{PSO}(n,1), \mathbb{H}^n)$$ and $$(\textrm{PO}(n+1),\mathbb{R}P^n)$$. The jist of the argument is as follows: consider the family of quadratic forms $$B_t=t(x_1^2+x_2^2+\cdots + x_n^2)-x_{n+1}^2$$. Then if $$t<0$$, $$B_t$$ has signature $$(n,1)$$, if $$t=0$$ the form is degenerate and if $$t>0$$ it has signature $$(n+1,0)$$. For the case $$t>0$$ we can define $$X_t\subset \mathbb{R}P^n$$ to be $$X_t=\{[v]\in\mathbb{R}P^n: v^TB_tV<0\}$$ and $$G_t=\mathrm{PSO}(B_t)$$. Then clearly when $$t=1$$ we have $$(G_1,X_1)=(\textrm{PSO}(n,1),\mathbb{H}^n)$$ and if we let $$C_t$$ be the diagonal matrix with the first $$n$$ entries equal to $$\sqrt{t}$$ and the last equal to $$1$$, we have $$C_tX_t=X_1$$ and $$C_tG_tC_t^{-1}=\textrm{PSO}(n,1)$$. Thus to show that Euclidean geometry is a conjugacy limit of hyperbolic geometry, it suffices to understand what happens to $$G_t$$ as $$t\to 0$$.

To do so, we will study their Lie algebras, and show that as $$t\to 0$$ the Lie algebra of $$G_t$$ goes to the Lie algebra of $$\textrm{Isom}(\mathbb{E}^n)$$. After a series of simple computations we have the following description of the Lie algebra $$\mathfrak{pso}(B_t)=\left\{\left(\begin{array}{cc}M&\frac{v}{\sqrt{t}}\\\sqrt{t}v^T&0\end{array}\right): M\in\mathfrak{so}(n), v\in\mathbb{R}^n\right\}$$. Then as $$t\to 0$$ this becomes

$$\left\{\left(\begin{array}{cc}M&w\\0 &0\end{array}\right): M\in\mathfrak{so}(n),w\in\mathbb{R}^n\right\}$$ which is the Lie algebra to the group of Euclidean isometries, as claimed.

A similar story can be told for $$t>0$$, and so we have Euclidean geometry arising as a conjugacy limit of both Hyperbolic and Elliptic.

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