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Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are related to $A$-polynomials and thus to various conjectures concerning asymptotics of quantum invariants such as the AJ-conjecture.

Varieties of representations and characters of surface groups in algebraic groups have also been intensively studied, this being a survey.

In recent years, there have been a large number of papers on varieties of representations and characters of $3$-manifold groups in $SL(n,\mathbb{C})$ for $n>2$, including the development of various tools to compute invariants associated to such varieties, e.g. work of Garoufalidis, Thurston, Zickert, and Goerner.

I have leafed through various of these papers on various occasions and heard some talks, but although understanding any sorts of representations of $3$--manifold groups is surely a worthy goal, as is generalizing results known for $n=2$ or for surface groups, I haven't really understood what motivates this direction of research or what people are aiming to achieve by studying such representations.

Question: What is the motivation for the study of representations of 3-manifold groups into $SL(n,\mathbb{C})$ for $n>2$?

For example, is it expected that such representations should provide tools to identify geometric structures on manifolds? Are they expected to offer insight into conjectures concerning asymptotics of quantum invariants? Or maybe something else?

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    $\begingroup$ If you had asked this question solely about surfaces (rather then 3-manifolds), I would have answered by pointing to the beautiful theorem of Hitchen, and the subsequent literature on the Hitchen component and associated geometric structures on surfaces and on bundles over surfaces. $\endgroup$ – Lee Mosher Mar 27 '16 at 16:31
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    $\begingroup$ Hitchin's theorem's even more beautiful :) $\endgroup$ – YCor Mar 28 '16 at 0:27
  • $\begingroup$ One thing you could say is that a homomorphism $\pi_1(M) \to SL(n,\mathbb{C})$ is the same as a flat connection on a principal $SL(n,\mathbb{C})$-bundle over $M$. Flat connections are important in physics? $\endgroup$ – Daniel Barter Apr 26 '16 at 20:20
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The simplest answer is that the universal cover together with the deck group action contain a lot of information about the manifold, and the representations of the group provide one way to extract it.

Representation of the group correspond to either locally constant sheaves or flat connections.

The acyclic representations, i.e., those for which the cohomology of the associated locally constant sheaf is trivial are particularly interesting. Weighted counts of such representations typically yield interesting invariants of the manifold. The Reidemeister torsion is a weighted count of acyclic $\mathbb{C}^*$-representations while the Casson invariant is a signed count of acyclic $SU(2)$-representations.

The space of representations into $SL(n,\mathbb{C})$ is naturally an algebraic variety equipped with an additional rich structure which can conceivably be used to produce invariants of the original manifold.

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  • $\begingroup$ Thanks! So this answer basically is that representations of 3-manifold groups into any group are interesting, and $SL(n,\mathbb{C})$ is just a class of groups such that (whatever version of) the algebraic variety of representations into it is relatively tractable? $\endgroup$ – Daniel Moskovich Mar 27 '16 at 17:58
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    $\begingroup$ That's basically the gist of it. You also need to remember that a representation of a fundamental group is a richer object object than a representation of an abstract group since it comes with a locally constant sheaf. $\endgroup$ – Liviu Nicolaescu Mar 28 '16 at 0:56
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    $\begingroup$ So the answer is that you might hope to generalize well known existing invariants? It would be nice to know what people then hope to do with those invariants. Since geometrization provides a very nice complete set of invariants for any 3-manifold, the business of finding new invariants for their own sake seems poorly motivated to me. $\endgroup$ – HJRW Apr 22 '16 at 16:24
  • $\begingroup$ What is the nice complete set of invariants supplied by the geometrization conjecture? What about the classification of knots and links? They are determined by their complements which are $3$-manifolds with boundary. Deciding whether two knots are isotopic is still a very difficult problem. $\endgroup$ – Liviu Nicolaescu Apr 22 '16 at 16:42
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    $\begingroup$ One last comment. You might also be interested in the discussion under this blog post: ldtopology.wordpress.com/2013/04/23/… . $\endgroup$ – HJRW Apr 26 '16 at 20:08
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Three manifold groups are quite special and their representation varieties have more structure than that of a random group. Casson's view point is helpful to see the point. The Heegaard decompoistion of a three manifold $Y= H_1\cup_\Sigma H_2$ means that the fundamental group of a three manifold has a quite special presentation. There is pushout diagram, \begin{eqnarray} \pi_1(\Sigma) & \to & \pi_1(H_1)\\ \downarrow & & \downarrow \\ \pi_1(H_2) & \to & \pi_1(Y) \end{eqnarray} and hence maps in the opposite direction on representation varieties.

\begin{eqnarray} R_G(\Sigma) & \leftarrow & R_G(H_1)\\ \uparrow & & \uparrow \\ R_G(H_2) & \leftarrow & R_G(Y) \end{eqnarray} The representation variety $R_G(\Sigma)$ is a symplectic manifold when $G$ is compact Lie group (thanks to Goldman and Atiyah-Bott) and gets a Kahler structure once metric is chosen on $\Sigma$. If $G$ is the complexification of a compact Lie group and a metric on $\Sigma$ is choose $R_G(\Sigma)$ is Hyperkahler manifold. (It is a slight lie (small l ;-) that these are manifolds.)
The maps $R_G(H_i) \to R_G(\Sigma)$ are injective and images (to the extent they are manifolds) are Lagrangian in case $G$ is compact and complex Lagrangian when $G$ is the complexification of a compact group. The intersection $$ R_G(Y)=R_G(H_1)\cap R_G(H_2) \subset R_G(\Sigma) $$ is then much more special, so in the compact case lies in the setting of Lagrangian Floer homology and the in the complex case in the setting is for example discussed recently by Witten and Haydys (see for example arXiv: 1010.2353 )

Related to this point of view is the fact that representation varieties of three manifolds are the set of critical points of a Chern-Simons functional on a suitable space of gauge equivalence classes of connections.

Apologies for the inaccuracies above due to haste and lazy typesetting.

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Here is another (very recent) motivation which might be more specific.

First, one application of the studying the $SL(2,\mathbb{C})$ character variety is that it detects some essential surfaces (i.e. surfaces that are incompressible and boundary incompressible). However the $SL(2,\mathbb{C})$ character variety is known not to detect all boundary slopes by the following:

Eric Chesebro and Stephan Tillmann, MR 2395254 Not all boundary slopes are strongly detected by the character variety, Comm. Anal. Geom. 15 (2007), no. 4, 695--723. (also https://arxiv.org/abs/math/0510418)

However, recently Friedl, Kitayama, and Nagel have announced that in fact all essential surfaces can be detected using $SL(n,\mathbb{C})$ character varieties.

Stefan Friedl, Takahiro Kitayama, and Matthias Nagel, Representation varieties detect essential surfaces http://arxiv.org/abs/1604.00584

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I would argue, as I do in the introduction of many of my papers, that the study of Betti Moduli Spaces $$\mathrm{Hom}(\Gamma, G)//G$$ for finitely generated $\Gamma$ and complex reductive $G$ is interesting in its own right, but that was not your question.

So with respect to 3-manifold groups in particular, let me quickly note a few people's work:

  1. Adam Sikora's work on higher Skein Theory and quantization: arXiv

  2. Hans Boden and Eric Harper's work on higher Casson Invariants: arXiv

  3. David Baraglia and Laura Schaposnik's work on Higgs bundles and branes: arXiv

Also, deformations of $(G,X)$-structures on 3-manifolds motivates studying these spaces. For example, $\mathbb{RP}^3$-manifolds; all of Thurston's eight geometries admit $\mathbb{RP}^3$-structures.

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