Groups acting on 3-manifolds

 In this post I would like to blog about two very nice results of John Pardon on groups acting on 3-manifolds. In spite of how natural they seem, they require a fair amount of technical ingenuity, which Pardon has impressively managed to put together so that we don't have to, and I will be somewhat brief on the details. 

The Hilbert-Smith conjecture claims the following: Let $G$ be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold $X$. Then $G$ is isomorphic to a Lie group. This is a generalisation of Hilbert's fifth problem and has been reduced to the case of $\mathbb{Z}_p$; see Tao's blog post for an exposition. I learned recently that John Pardon proved this for all manifolds of dimension at most 3, and found a nice sketch of the argument here. 

One of the main tools involved is the construction of a lattice of incompressible surfaces in a 3-manifold, where here a lattice is meant in the sense of a poset where every finite set of elements have a least upper bound. Before we define this, we discuss the analogous lattice for surfaces where it is hopefully easier to visualise.

Let \( F \) be a surface (without boundary, possibly non-compact). We denote by \( \mathcal{L}(F) \) the set of isotopy classes of codimension zero submanifolds-with-boundary \( A \subseteq F \) for which \( \partial A \subseteq F \) is a compact multi-curve, such that neither \( A \) nor \( A^c := F \setminus A \) have any components diffeomorphic to \( D^2 \), \( S^1 \times I \), or \( S^1 \times I \). This implies that \( \partial A \) consists of pairwise non-isotopic essential curves on \( F \). The isotopy class of any such \( A \subseteq F \) is contractible.

There is a partial order on \( \mathcal{L}(F) \) by inclusion. Namely, \( \mathfrak{A} \leq \mathfrak{A}' \) iff there are representatives \( A, A' \subseteq F \) of \( \mathfrak{A} \) and \( \mathfrak{A}' \) with \( A \subseteq A' \). Obviously \( A \mapsto A^c \) is an order reversing involution of \( \mathcal{L}(F) \).

Equipped with this partial order, \( \mathcal{L}(F) \) is a lattice, namely every finite subset \( S \subseteq \mathcal{L}(F) \) has a least upper bound. To see this, fix a hyperbolic metric on \( F \) with no parabolics, so every isotopy class of simple closed curve has a unique geodesic representative, which is length minimizing. Now every isotopy class \( \mathfrak{A} \in \mathcal{L}(F) \) has a unique representative \( A \subseteq F \) whose boundary consists of a disjoint union of length minimizing geodesics. These representatives simultaneously realize all order relations, in the sense that for such \( A, A' \subseteq F \), if \([A] \leq [A']\) then \( A \subseteq A' \). This implies the lattice property as follows. For finite \( S \subseteq \mathcal{L}(F) \), consider the unique representatives \( A_s \subseteq F \) whose boundaries \( \partial A_s \) are disjoint unions of length minimizing geodesics. The union \( \bigcup_{s \in S} A_s \subseteq F \) will not have any \( D^2 \), \( S^1 \times I \), or \( S^1 \times I \) components, however its complement may have such. Adding in these disallowed components produces the desired least upper bound, due to the fact that any \( \mathfrak{A}' \in \mathcal{L}(F) \) with \( \mathfrak{A}' \geq [A_s] \) for all \( s \in S \) has a representative \( A' \) with \( A' \supseteq A_s \) for all \( s \in S \).


Now we can treat 3-manifolds. Let \( M \) be a \( P^2 \)-irreducible three-manifold-with-boundary, and let \( A \subseteq \partial M \) be a codimension zero submanifold-with-boundary representing an element of \( \mathcal{L}(\partial M) \). We will define a lattice \( \mathcal{L}(M; A) \).

Throughout this subsection, the notation \( B \subseteq M \) (or its decorations such as \( B', B_1, \bar{B}, \) etc.) will always indicate a codimension zero submanifold-with-boundary such that \( B \cap \partial M = A \) and \( \partial B \subseteq M \) is a compact properly embedded surface-with-boundary. (Here and below we will, contrary to the usual meaning of \( \partial \), use \( \partial B \) to denote the boundary of \( B \) as a subset of \( M \) in the sense of point set topology.)

Given any \( B \subseteq M \), we may perform any of the following operations:

•  Removal of a neighbourhood of a disk \( (D^2, \partial D^2) \hookrightarrow (B, \partial B) \) with essential boundary.
• Removal of a component of \( (B, \partial B) \) which is diffeomorphic to \( (B^3, S^2) \).
• Removal of a component of \( (B, \partial B) \) which is diffeomorphic to \( (F \times I, F \times \partial I) \) for a closed surface \( F \).

Such an operation will be called a compression, and $B$ is called incompressible if it admits no such operations. If $B$ is incompressible, then it is $P^2$-irreducible (since $M$ is $P^2$-irreducible and $B^3$ contains no closed incompressible surfaces). As mentioned above, the main result needed is that this is a lattice. This setup is more general than in the linked blog post above, because Pardon uses this souped up version to resolve another longstanding open problem which we now describe.

Every continuous finite group action on a manifold of dimension \( \leq 2 \) is conjugate to a smooth action. Bing, Montgomery--Zippin, and Alford have constructed examples of continuous finite group actions on 3-manifolds which aren't conjugate to smooth actions, but are defined as uniform limits of smooth actions. Pardon managed to show that this is the only way to construct non-smooth actions. Specifically, he proved

Theorem: Every continuous action \( \varphi : G \curvearrowright M \) of a finite group on a smooth three-manifold is a uniform limit of smooth actions \( \tilde{\varphi} : G \curvearrowright M \). If \( \varphi \) is smooth over \( \operatorname{Nbd} K \) for \( K \subseteq M \) closed and \( \varphi(G) \)-invariant, then we may take \( \tilde{\varphi} = \varphi \) over \( \operatorname{Nbd} K \).

This theorem is in some sense sharp, as shown by the following remark that Pardon attributes to Bestvina:

Remark: In higher dimensions, there exist (even free) finite group actions which are not uniformly approximable by smooth actions. Indeed, let \( M \) be a topological manifold of dimension \( \geq 5 \) which admits no smooth structure but which admits a finite cover \( \tilde{M} \to M \) with \( \tilde{M} \) admitting a smooth structure (e.g. \( M \) could be a non-smoothable fake real projective space ). By passing to a further cover, we may assume that \( \tilde{M} \to M \) is Galois, say with Galois group \( G \). Choosing arbitrarily a smooth structure on \( \tilde{M} \), we claim that the continuous free action \( \varphi : G \curvearrowright \tilde{M} \) is not uniformly approximable by smooth actions. Indeed, for any action \( \varphi' : G \curvearrowright \tilde{M} \) sufficiently uniformly close to \( \varphi \), the quotients \( M = \tilde{M}/\varphi(G) \) and \( \tilde{M}/\varphi'(G) \) are homeomorphic by the Chapman--Ferry \( \alpha \)-approximation theorem. Hence \( \varphi' \) cannot be smooth, as otherwise we would obtain a smooth structure on \( M \), which we have assumed does not exist.

Here is Pardon's sketch of the proof of the theorem, starting with the case of free actions. If \(\varphi : G \curvearrowright M\) is free, then the quotient space \(M/G\) is a topological manifold. Bing and  
Moise proved that every 3-manifold admits a smooth structure, so there exists a smooth structure on \(M/G\), which we can pull back to a smooth structure on \(M\) (call it \(M^s\)) with respect to which \(\varphi\) is smooth. Now the identity map id : \(M \to M^s\) is a homeomorphism between smooth three-manifolds, and Bing and Moise further proved that any homeomorphism between smooth three-manifolds can be uniformly approximated by diffeomorphisms. Denoting by \(\alpha : M \to M^s\) such a diffeomorphism, we conclude that the conjugated action \(\alpha^{-1}\varphi\alpha : G \curvearrowright M\) is smooth and uniformly close to \(\varphi\). 

To treat more general actions \(\varphi : G \curvearrowright M\), we need some understanding of which subsets of \(M\) can occur as the fixed points of the action of \(G\) or of one of its subgroups. Smith theory concerns precisely this question, and provides that for any homeomorphism \(g\) of prime order \(p\) of a topological three-manifold \(M\), the fixed set \(M^g\) is a topological manifold (of possibly varying dimension and possibly wildly embedded inside \(M\)). Writing  

\[M^g = M_{(0)}^g \sqcup M_{(1)}^g \sqcup M_{(2)}^g \sqcup M_{(3)}^g\]

for the decomposition of \(M^g\) by dimension, we furthermore have that \(M_{(2)}^g\) can be non-empty  only when \(p = 2\) and \(g\) reverses orientation near \(M_{(2)}^g\).

action of \( G \) such that all triple intersections are empty (\( U_i \cap U_j \cap U_k = \emptyset \) for distinct \( i, j, k \)).  
Using the lattice of incompressible surfaces, we now find properly embedded incompressible surfaces \( F_{ij} \subseteq U_i \cap U_j \) (separating the \( U_i \) end and the \( U_j \) end) which are \( G \)-invariant up to isotopy. 

These surfaces \( F_{ij} \) divide \( M_0 \) into pieces \( N_i \) (each a compact three-manifold-with-boundary), and \( G \) acts on \( \bigsqcup_i N_i \) up to homotopy. Finally, Pardon explains that these homotopy actions can be upgraded to strict actions. The resulting smooth action of \( G \) on \( M \) can be made arbitrarily close to the original action in the uniform topology by taking the neighbourhood \( M_0 \) and the open sets \( U_i \) to be sufficiently small.

A lot of the technical work goes into the upgrading homotopy actions to strict actions, which he calls Nielsen realisation on 3-manifolds after the analogous problem for surfaces that was solved by Kerckhoff. Specifically, Kerckhoff proved that a finite subgroup of the mapping class group in fact does come from a finite subgroup of the group of homeomorphisms. Pardon does all Seifert-fibred manifolds and hyperbolic manifolds, then argues that he can glue these together along the JSJ decomposition. This is quite demanding, but for instances the case of a hyperbolic piece is mostly Mostow rigidity.