4 Covering theory

This section recounts the theory of universal covers for “nice” topological spaces.

In particular, we aim to understand (and program) how the fundamental groups of closed hyperbolic \(3\)-manifolds can be represented by matrix subgroups of the isometry group of \(\mathbb H^3\), and how such representations can be visualized as “lattice tilings” of \(\mathbb H^3\).

  1. Review the basics of covering theory.
  2. We revisit the proof of the Švarc–Milnor lemma (Loh, 5.4), which gives an explicit construction of the quasi-isometric embedding of the Cayley graph of a fundamental group into the universal cover.

To obtain the quasi-isometry constants of the embedding \(\Delta(\pi_1(\text{Weeks manifold}), \{a, b\})\), we:

  1. Compute the generating set \(S\) of \(\pi_1(\text{Weeks manifold})\) according to the Švarc–Milnor lemma.
  2. Compute the quasi-isometry constants of the embedding \(\Delta(\pi_1(\text{Weeks manifold}), S)\).
  3. Use the fact that \(\Delta(\pi_1(\text{Weeks manifold}), \{a, b\})\) is quasi-isometric to \(\Delta(\pi_1(\text{Weeks manifold}), S)\).

Topological preliminaries

See Loh (A.1.2) and Munkres for more details. Here we recount the basics of covering theory.

  • Every “nice” topological space has a universal cover \(\widetilde X\) which is simply connected.
  • The space \(\widetilde X\) admits a free, properly discontinuous group action of \(\pi_1(X, x_0)\).
    • There is a homeomorphism from \(\widetilde X\) modulo \(\pi_1(X, x_0)\) to \(X\) by sending \(x_0\) to the class of \(\widetilde x_0\). In a sense \(X\) is a “representative pancake” in \(\widetilde X\).
  • If \(X\) is a metric space, then \(\widetilde X\) carries the induced path-metric and the action of \(\pi_1(X, x_0)\) is isometric, so fundamental groups are subgroups of the isometry group of the universal cover.

Some more results about hyperbolic spaces:

  1. Using the hyperboloid model of \(\mathbb H^3\) as the \(x_0>0\) slice of \(\mathbb R^4\) under the contour equation \(-x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1\), we obtain \[ \mathrm{Isom}\, \mathbb H^3 = O^+(3, 1) \] where \(O^+(3, 1)\) is the group of proper orthochronous Lorentz transformations which keep the upper-sheet of the hyperboloid invariant.
  2. A standard result has \(\mathrm{SO}^+(3, 1)\cong \mathrm{PSL}(2, \mathbb C)\), i.e. there is a \(2\)-to-\(1\) covering \(\mathrm{SL}(2, \mathbb C)\to \mathrm{SO}^+(3, 1)\).
  3. Need to double-check: for orientable closed manifolds (e.g. the Weeks manifold?), the deck transformations are orientation-preserving, so \(\pi_1(\text{Weeks manifold}) \subset \mathrm{SO}^+(3, 1)\).

Švarc–Milnor lemma

Theorem 4.1 (Švarc–Milnor lemma) Given the group action of \(G\) on a non-empty metric space \((X, d)\) such that \(\forall g: x\mapsto g\cdot x\) is an isometry. Further suppose that \(X\) is \((c, b)\)-quasi-geodesic and that there is a subset \(B\subset X\) such that:

  1. \(\mathrm{diam}\, B\) is finite.
  2. The \(G\)-translates of \(B\) cover \(X\) (not necessarily disjoint).
  3. Let \(B'\) be the \(2b\)-thickening of \(B\), the set \(S = \{g\in G \, | \, g\cdot B'\cap B'\neq \emptyset\}\) is finite.
    • The generating set \(S\) is geometrically the neighboring set which sends the thickenedfundamental domain \(B'\) to overlapping neighboring translations.

Then the following holds:

  1. \(G\) is finitely generated by \(S\).
  2. Fixing any basepoint \(x\in X\), the Cayley graph of \(G\)
    is quasi-isometric to \(X\) by the map \[ G\xrightarrow {g\mapsto g\cdot x} X \]
  3. In particular, we obtain the following quasi-isometric bound: \[ \dfrac b {c^2} d_S(e, g) - b \left(\dfrac 1 {c^2} - 1\right) \leq d(x, g\cdot x) \leq 2(\mathrm{diam}\, B + 2b) \cdot d_S(e, g) \tag{4.1} \]

Proof: The proof proceeds as follows:

  1. Proving that \(S\) generates \(G\): given \(g\in G\), connect \(x, g\cdot x\) by a quasi-geodesic, then divide the geodesic into sufficiently fine intervals with endpoints covered by \(g_j\) such that \(g_{j-1}\cdot B'\cap g_j\cdot B'\) is nonempty, so that \(g_{j-1}^{-1}g_j\) is in the neighboring set.
  2. Proving that \(g\mapsto g\cdot x\) is a quasi-dense: follows from the tiling of \(X\) by \(G\)-translates of \(B\).
  3. Proving that \(g\mapsto g\cdot x\) is a quasi-isometric embedding:
    • Use the lower bound provided by the quasi-geodesicity.
    • Upper-bound using the fact that subsequent endpoints \((x_j)\) cannot be too far apart.

Fix \(g\in G\) for which we wish to show that \(g\in \langle S\rangle_G\). Choose any \(x\in B\) and construct a \((c, b)\)-quasi-geodesic \(\gamma:[0, L]\to X\) connecting \(x\) and ending in \(g\cdot x\).

  • Divide up the interval \([0, L]\) into sufficiently fine intervals such that \(x_0=x, x_n=g\cdot x\), and that \(d(x_{j-1}, x_j) \leq c\cdot |t_{j-1}-t_j|_{\leq b/c} + b \leq 2b\).

We proceed to show that \(\forall j\in \{1, \dots, n\}\), \[ s_j = g_{j-1}^{-1} g_j \in S = \{g\in G \, | \, g\cdot B'\cap B'\neq \emptyset\} \] To show this, \(x_j\in B_{2b}(g_{j-1}\cdot B) = g_{j-1}\cdot B_{2b}(B) = g_{j-1}\cdot B'\). Similarly, \(x_j\in g_j\cdot B\subset g_j\cdot B'\) so \(g_{j-1}\cdot B'\cap g_j\cdot B' \supset \{x_j\}\) and \(s_j=g_{j-1}^{-1}\cdot g_j\in S\). Then \[ g = g_n = g_{n-1} g_{n-1}^{-1} g_n = g_{n-1}s_n = g_{n-2}s_{n-1}s_n = \cdots = s_1\cdots s_n \in \langle S\rangle \] To show that \(\varphi:g\mapsto g\cdot x\) is quasi-dense, simply note that the quasi-dense constant is bounded by \(\mathrm{diam}\, B\) since the \(g\)-translates of \(B\) cover \(X\). To next show that \(\varphi\) is a quasi-isometric embedding, using the isometric condition \(g(\varphi\, g, \varphi\, h) = d(e, \varphi(g^{-1}h))\) and \(d_S(g, h) = d_S(e, g^{-1}h)\) we obtain

  1. Lower bound \(d(x, g\cdot x)\) in terms of \(d_S(e, g)\): let \(\gamma\) be a quasi-geodesic from \(x\) to \(g\cdot x\) as in the first part, then noting that \(d_S(e, g) \leq n\) (since \(g=s_1\dots s_n\)) we obtain \[\begin{align*} d(x, g\cdot x) &= d(\gamma_0, \gamma_L) \geq \dfrac 1 c L - b \geq \dfrac 1 c \cdot \dfrac{b}{c}(n-1) - b \\ &= \dfrac{b}{c^2} \cdot n - \dfrac{b}{c^2} - b \geq \dfrac b {c^2}d_S(e, g) - \dfrac{b}{c^2} - b \end{align*}\]
  2. Upper bound of \(d(x, g\cdot x)\) in terms of \(d_S(e, g)\): let \(d_S(e, g)=n\) so that \(g=s_1\dots s_n\) with \(s_j\in S\cup S^{-1}\). Then using the triangle inequality, the isometric condition \(d(g\cdot x, g\cdot y)=d(x, y)\), we obtain \[\begin{align*} d(x, g\cdot x) &\leq d(x, s_1\cdot x) + d(s_1\cdot x, s_1\cdot s_2\cdot x) + \cdots + d(s_1\cdots s_{n-1}\cdot x, s_1\cdots s_n\cdot x) \\ &= d(x, s_1\cdot x) + d(x, s_2\cdot x) + \dots + d(x, s_n\cdot x) \\ &\leq n\cdot 2(\mathrm{diam}\, B + 2b) \leq 2(\mathrm{diam}\, B + 2b) \cdot d_S(e, g) \end{align*}\] In the last line we used the fact that \(\forall x, s\in S = \{g\in G \, | \, g\cdot B'\cap B'\neq \emptyset\}\), we have \(d(x, s\cdot x) \leq 2(\mathrm{diam}\, B + 2b)\): let \(y\in s\cdot B'\cap B'\), then \[ d(x, s\cdot x) \leq d(x, y) + d(y, s\cdot x) \leq (\mathrm{diam}\, B + 2b) + (\mathrm{diam}\, B + 2b) \]

Corollary 4.1 (fundamental groups and universal covers) Given a compact, boundaryless Riemannian manifold \(M\) with universal cover \(\widetilde M\), the fundamental group \(\pi_1(M)\) is finitely-generated and for every basepoint \(x\in \widetilde M\), the map \[ \pi_1(M)\xrightarrow {g\mapsto g\cdot x} \widetilde M \] given by the action of the fundamental group on \(\widetilde M\) by deck transformations (analogous to reshuffling pancakes) is a quasi-isometry with bound \[ b \cdot d_S(e, g) \leq d(x, g\cdot x) \leq 2(\mathrm{diam}\, B + 2b) \cdot d_S(e, g) \tag{4.2} \] Note that varying \(b\) will vary the generating set S.

Proof: The action of deck transformations for closed Riemannian manifolds is isometric, proper, and cocompact. Also note that \(\widetilde M\) is \((1, b)\)-quasi-geodesic for every \(b\in \mathbb R_{>0}\). Given such action, we simply choose \(B=M\) in theorem 4.1. To obtain the claimed bound, substitute \(c=1\) in (4.1).