Faithful representations of surface groups

Given the success of character theoretic methods in the theory of finite groups, one might very reasonably ask whether this extends to infinite groups as well. The answer is yes, but more nuance is needed when asking these questions. In the case of finite groups character theory is a rather powerful invariant for classifying representations over fields, even in positive characteristic, but when groups are infinite this tends to miss out a lot of information, and in general much less is possible. For instance, classifying all irreducible representations by their characters seems to be a hopeless task. Hence we tend to look at the space of all possible representations, which will be called the representation variety, endow it with a geometric structure, because it naturally comes with one and doing so is convenient, and then instead look at particular subsets of the representation variety depending on the context. 

Let $G$ be a discrete group with a finite generating set of size $k$. Suppose we want to consider the space of all representations $G \to \mathrm{GL}_n(\mathbb{C})$. A map from $G$ to $\mathrm{GL}_n(\mathbb{C})$ is uniquely specified by a choice of $k$ matrices, one for each generator, such that these matrices satisfy the relations in $G$. These relations are polynomials in the entries of the matrices, so the relations cut out an ideal in the variety $\mathrm{GL}_n(\mathbb{C})^k$, and in fact it is defined by finitely many relations by Hilbert's basis theorem. Thus, the space of representations, denoted $\mathrm{Hom}(G,\mathrm{GL}_n(\mathbb{C}))$, naturally carries the structure of an affine variety, as well as the subspace topology inherited from $\mathrm{GL}_n(\mathbb{C})^k$. We call the respective topologies the Zariski and analytic topologies. For instance, when $G=F_k$, a free group of rank $k$, $\mathrm{Hom}(G,\mathrm{GL}_n(\mathbb{C}))=\mathrm{GL}_n(\mathbb{C})^k$ since there are no relations one needs to take care of.

This however, is easily generalised. I picked $\mathrm{GL}_n(\mathbb{C})$ because the easiest case in representation theory of finite groups is over an algebraically closed field of characteristic 0, but one can take any choice of Lie group and get a perfectly sensible object. For $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$ and Lie group $H$ over $\mathbb{K}$, the set $\mathrm{Hom}(G,H)$ is naturally a $\mathbb{K}$--algebraic set, called the $H$ representation variety of $G$. The study of these objects goes by the name of (higher) Teichmuller theory. 

As a geometric group theorist, after knowing what happens in the case of free groups and finite groups, the next thing I want to know is what happens for surface groups. So, let $\pi$ be the fundamental group of a closed oriented surface $\Sigma$ of genus $g \geq 2$. Already this gets quite interesting: it doesn't seem obvious how to study when matrices satisfy the defining relation (a product of commutators of the generators), so one ends up using geometric techniques instead: for instance, a hyperbolic manifold is given by a faithful map from its fundamental group into the corresponding Lie group, such that the image has finite covolume. This is a very useful alternative perspective. Surfaces are lattices in $\mathrm{PSL}_2(\mathbb{R})$, and for a long time it was unknown how surfaces sit in 3-manifolds, especially hyperbolic ones (lattices in $\mathrm{PSL}_2(\mathbb{C})$. This was known as the virtual Haken conjecture, which was spectacularly proved by the work of many people, especially Kahn-Markovic and Agol. Kahn-Markovic prove, in particular, that hyperbolic 3-maifolds contain loads and loads of $\pi_1$-injective immersed surfaces. Long before this, however, people wanted to know that in general there were loads of surface subgroups in $\mathrm{PSL}_2(\mathbb{K})$, so they studied the corresponding representation variety.

Given a representation $\phi\colon\thinspace \pi \rightarrow \mathrm{PSL}_2(\mathbb{K})$, Milnor proved that there is an associated flat principal $\mathrm{PSL}_2(\mathbb{K})$--bundle over $\Sigma$. The only obstruction to building a cross-section of this bundle is a class $o_2(\phi)$ in $H^2(\pi; \pi_1(\mathrm{PSL}_2(\mathbb{K}))) \cong \pi_1(\mathrm{PSL}_2(\mathbb{K}))$. When $\mathbb{K}=\mathbb{R}$, $o_2(\phi)$ is the Euler number $e(\phi)$---so called as it is the Euler number of the associated $\mathbb R \mathrm P^1$--bundle over $\Sigma$---and when $\mathbb{K}=\mathbb{C}$, $o_2(\phi)$ is the second Stiefel-Whitney class $w_2(\phi)$---as it is the second Stiefel-Whitney class of the associated $\mathbb H^3$--bundle over $\Sigma$. The second Stiefel-Whitney class of a real representation is the Euler number modulo two.

Goldman proved that the topological components of $\mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{C}))$ are $w_2^{-1}(i)$ for $i=0,1$ and those of $\mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{R}))$ are $e^{-1}(n)$ for $2-2g \leq n \leq 2g-2$, i.e. the above invariants are in fact complete invariants for determining which connected component of the representation variety it lives in. Furthermore, each representation variety has two irreducible components, corresponding to the two possible Stiefel-Whitney classes.

While these results are impressive, this still didn't show that there were actual surface subgroups. A little surprisingly, it took until 2004 for DeBlois and Kent to offer a proof of the following:

Theorem 1: For $\mathbb{K} = \mathbb{C}$ or $\mathbb{R}$, the set of faithful representations is dense in $\mathrm{Hom}(\pi,\mathrm{PSL}_2(\mathbb{K}))$ equipped with its analytic topology.

We'll give a sketch of the proof. However, in addition to the niceness of the theorem, another reason I decided to write this post is because the proof draws attention to a certain subtlety when dealing with algebraic geometry over the reals which I think people should be aware of. We discuss this first.

Density in the analytic topology

Here is an example that shows that the complement of a proper subvariety of an irreducible real algebraic variety is not always dense. The two-variable polynomial \[ p(x,y) = y^2 - x^2(x-1) \] defines a typical example of a singular elliptic curve, which in this case is singular at the origin, and is irreducible, so the set \[ V(p) = \{ (x,y) \in \mathbb{R}^2\ |\ p(x,y)=0 \} \] is an irreducible real algebraic variety.  The points $(0,0)$ and $(1,0)$ are the only elements of $V(p)$ with $y$-coordinate equal to $0$, and if $(x,y) \in V(p)$ with $y \neq 0$, then $y^2 >0$, which implies that $x>1$.  Hence $(0,0)$ is an isolated point of $V(p)$ in the analytic topology, even though $V(p)$ is one dimensional. It is also a proper subvariety, which, by the above, is not approached by points in its complement.

In this example, the isolated point is the origin, which is not a smooth point of the variety. The issue here is really the same issue that pops up whenever working with fields that aren't algebraically closed: i.e. the points which 'should be' there are missing. In this case it so happens that the approximating points live in the complex world. The following fact shows that the situation at smooth points is analogous to the complex case.

Lemma 2: Let $X \subset \mathbb{R}^n$ be a real algebraic variety of dimension $k$, and let $x \in X$ be a smooth point.  Let $Y \subset X$ be a subvariety of dimension $l < k$ with $x \in Y$. Then $x$ is approached by a sequence in $X-Y$. 

Proof: The proof is by induction on $k$.

If $k=1$, then $l=0$.  Hence $Y$ consists of a finite collection of points.  Since $x$ is a smooth point of $X$, there is a chart $\phi\colon\thinspace (0,1) \rightarrow X$ around $x$. Now pick any sequence in $(0,1)$ that approaches $\phi^{-1}(x)$ and misses $\phi^{-1}(Y)$.  

Now suppose $k>1$.  For each irreducible component $Y_i$ of $Y$, Let $y_i \in Y_i-\{x\}$, and choose an affine hyperplane $A$ so that $x \in A$, $y_i \notin A$ for all $i$, and $A$ is transverse to $X$ at $x$.
Then $A \cap X$ is $(k-1)$--dimensional and no irreducible component of $Y$  is contained in $A \cap X$. So for each $i$, \[ \mbox{dim}\ Y_i \cap A \leq l-1 < k-1. \] Furthermore, $x$ is a smooth point of $X \cap A$ since the tangent space at $x$ is of the proper dimension, and the claim  follows by induction. $\blacksquare$

Now for a variety $X$ in which smooth points are dense, any point $x$ in a proper subvariety $Y$ is approached by smooth points of $X$, each of which is either in $X-Y$ or approached by a sequence in $X-Y$.  A diagonal argument gives a sequence in $X-Y$ approaching $x$.

In order to apply the results of these section to prove theorem 1 in the real case, we need to check an extra condition smooth points to be dense. Happily for us, Goldman proved exactly that the set of smooth points of $\mathrm{Hom}(\pi,\mathrm{PSL}_2(\mathbb{R}))$ is dense in $\mathrm{Hom}(\pi,\mathrm{PSL}_2(\mathbb{R}))$.

Proof of theorem 1

By some fairly explicit calculations, DeBlois-Kent prove the following result, which is the main tool needed for the proof of theorem 1:
Theorem 3: For any nontrivial $w \in \pi$, there is a representation $\phi\colon\thinspace \pi \rightarrow \mathrm{PSL}_2(\mathbb{R})$ such that $e(\phi)=3-2g$ and $\phi(w)$ is nontrivial.

We blackbox this and Goldman's results and use them to prove theorem 1.

Proof: Write \[ \mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{C}))= X_0 \cup X_1, \] where $X_i$ is the irreducible component consisting of representations with Stiefel-Whitney class $i$ for $i=0,1$. Similarly, write \[ \mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{R}))= Y_0 \cup Y_1, \] where $Y_i$ is the irreducible component consisting of representations with Euler number equal to $i$ modulo 2 for $i=0,1$.  Note that $Y_i \subset X_i$ under the natural inclusion $\mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{R})) \subset \mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{C}))$. For an element $w$ of $\pi$, let $X_w$ (respectively, $Y_w$) be the algebraic subset of $X_1$ ($Y_1$) consisting of representations killing $w$.  

By Theorem 3, if $w \in \pi$ is nontrivial then $X_w \subset X_1$ and $Y_w \subset Y_1$ are proper algebraic subsets. It is a standard fact about irreducible complex varieties that the complement of any proper subvariety is dense in the analytic topology. Appealing to the discussion above and Goldman's result on smooth points, the same is true of the real variety that we care about. Hence, $X_1-X_w$ is an open dense subset of $X_1$, and it follows that $Y_1-Y_w$ is an open dense set of $Y_1$. 

But the set of faithful representations in $Y_1$ is precisely \[ \bigcap_{1 \neq w \in \pi_1 \Sigma} Y_1 - Y_w \] and similarly for faithful representations in $X_1$. This is an intersection of open dense subsets, which by the Baire Category Theorem is dense.

This proves that the set of faithful representations is dense in the irreducible component of each representation variety corresponding to a nonzero Stiefel-Whitney class.  In the other component this is immediately evident by the above argument, since Fuchsian representations (discrete subgroups of $\mathrm{PSL}_2(\mathbb{R})$), for instance, are faithful with Stiefel-Whitney class 0. $\blacksquare$

In hindsight, the proof looks quite natural when one knows the correct algebraic geometry phenomena to be aware of. However, I don't really have a good feeling for how hard to find the proof of theorem 3 is, and that is probably where the real difficulty lies.