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I am familiar with the construction and general properties of trace fields for hyperbolic $3$-manifolds. But in that setting we use Moscow-Prasad rigidity to define these fields as manifold invariants. I want to understand how the trace field can be defined as a manifold invariant for a hyperbolic surface.

In particular, if $X$ is a complete, orientable, finite-volume, hyperbolic $3$-manifold, then by Mostow-Prasad Rigidity, the set of discrete faithful representations of $\pi_1(X)$ into $\mathrm{PSL}_2(\mathbb{C})$ makes up an isomorphism class of Kleinian groups. Since these isomorphisms are given by conjugation, which doesn't effect the trace, we can pick one of these Kleinian groups and use the traces in it to form the trace field as a manifold invariant.

Now let $S$ be a complete, orientable, finite-volume hyperbolic surface. Then $\pi_1(S)$ admits a Teichmüller space of discrete faithful representations into $\mathrm{PSL}_2(\mathbb{R})$ which are not necessarily isomorphic. How is the trace field defined, exactly, in this setting, as an invariant of $S$? Is it formed by all the traces from all the representations?

Also, a reference on this would be cool. I dug around for a while before posting this but could not find anything with enough detail.

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You are not going to get very much information by studying the set of traces of all elements of $S$ over all discrete faithful representations of $S$.

For example, consider any nontrivial, nonperipheral element $\gamma \in \pi_1(S)$ represented by a simple closed curve. Using Fenchel-Nielsen coordinates, given any positive length $\ell > 0$ I can find a complete, finite area hyperbolic structure on $S$ in which $\gamma$ has length $\ell$. Thus, given any $T > 2$, I can find a discrete, faithful, finite area representation of $\pi_1(S)$ in $\text{PSL}_2(\mathbb{R})$ for which $|\text{trace}(\gamma)|=T$. The set of absolute values of traces of all elements of all representations is therefore equal to the entire interval $(2,+\infty)$.

Instead, what people study is the trace field of a particular point in Teichmüller space. That's an interesting and informative thing to do.

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  • $\begingroup$ Okay thanks I see what you mean. I guess one easy way of getting a surface analog of what happens in 3D is to study the surfaces up to isometry instead of up to homeomorphism. But then my trace field is generally going to be transcendental right? I'm wondering now if there's a well known way of picking a "nice" point in the Teichmuller space of $S$ so that certain properties of the resulting field could still be indicative of the whole homeomorphism class. (I know this is more straightforward for non-compact arithmetic surfaces, and for triangle groups, but is there a general theory?) $\endgroup$ – j0equ1nn Nov 8 '16 at 20:48
  • $\begingroup$ There is certainly a large literature on arithemtic hyperbolic surfaces. Whether there is some kind of unifying general theory of trace fields of all hyperbolic surfaces, I do not know, but I suspect the arithemetic surface people could say something. $\endgroup$ – Lee Mosher Nov 8 '16 at 23:51
  • $\begingroup$ Great, much appreciated again Lee (you've helped me in the past with similar concepts on Mathoverflow). In fact I have what I need for my current project, so I'll store these questions for a future conversation. $\endgroup$ – j0equ1nn Nov 9 '16 at 3:17

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