I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to K_3^{\operatorname{ind}}(\mathbb{C})$ is surjective.

Question 0, a historical aside: Recently I grew a little uncertain about the correct attribution of this question/conjecture. I have seen this conjecture attributed to Bloch somewhere, and in Dupont's book "Scissors congruences, group homology and characteristic classes" it is attributed to Sah. What's the exact history?

Now for the actual mathematical questions. Recall that one can associate classes in $K_3$ (or versions of the Bloch group) to hyperbolic $3$-manifolds. Roughly, this is done by choosing a triangulation of the manifold $M$ by ideal tetrahedra, and the ideal tetrahedra naturally give classes in the scissors congruence (or pre-Bloch) group. There are several papers by Neumann and Yang about this. My main question is now: what, if anything, corresponds to the rigidity question for $K_3$ on the hyperbolic $3$-manifold side? More detailed questions are formulated below:

Question 1: What are possible reasons for believing or disbelieving the conjecture? I guess the rigidity of the Cheeger-Chern-Simons invariants is one reason for believing rigidity, maybe another is that we have no clue what else could be in $K_3^{\operatorname{ind}}(\mathbb{C})$? Any other reasons? Or maybe there is an argument why hyperbolic geometry can not possibly see non-rigidity of $K_3$? I would be mainly interested in intuition from the hyperbolic geometry side, about which I know almost nothing.

Question 2: Assume, just for the fun of it, that the conjecture is false, i.e., there are classes in $K_3^{\operatorname{ind}}(\mathbb{C})$ which do not come from $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})$. Would there be a hyperbolic geometry interpretation of these classes?

Something related to Question 2 was discussed in Ian Agol's answer to this MO-question. Apparently, one would not see the "new" classes as manifolds with strange triangulations, but rather the new classes would come from deformations of $\operatorname{SL}_2(\mathbb{C})$-representations of the fundamental group of $M$. The representations would correspond to flat rank $2$ vector bundles. Is it possible to say that a failure of the rigidity conjecture would imply the existence of strange deformations of flat rank $2$ vector bundles on hyperbolic $3$-manifolds?

Are there other things/objects in hyperbolic geometry that would deform in a strange way if the rigidity conjecture was false? I guess it is called rigidity conjecture for a reason.

Question 3: Have people considered a way of going from classes in the Bloch group to hyperbolic manifolds? An element of the Bloch group can be represented as a linear combination of ideal tetrahedra, but it is not obvious to me how I could get a hyperbolic $3$-manifold from that? Is the relation defining the Bloch group ($\sum x\wedge (1-x)=0$) enough to make sure that the ideal tetrahedra can be glued to a manifold in some way?

Provided Question 3 has a positive answer, then I would have a way of interpreting elements in $K_3^{\operatorname{ind}}(\mathbb{C})$ which do not come from $\overline{\mathbb{Q}}$ (assuming these exist). For simplicity, let $C$ be a smooth projective curve over $\overline{\mathbb{Q}}$. Then I would interpret elements in $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}}(C))$ as (linear combinations of) deformations of ideal tetrahedra with boundary points in $\overline{\mathbb{Q}}$ with parameter space some open subcurve of $C$. If the simplices can be glued, that would provide a hyperbolic $3$-manifold together with a deformation of a triangulation (with parameter space a subcurve of $C$). The fact that the corresponding element in $K_3$ does not come from $\overline{\mathbb{Q}}$ would say that the deformation of the triangulation can not be made constant by the obvious operations on ideal tetrahedra. Is it true that I can view non-rigid elements in $K_3$ as deformations of ideal triangulations?

Question 4: Now this is only meaningful if the questions above have a reasonably positive answer. Assuming the failure of the rigidity conjecture, and assuming that it is possible to represent non-rigid elements by deformations of hyperbolic-geometric objects (vector bundles or triangulations or some such thing), would there be invariants (other than their classes in $K_3$) that one could use to show that such objects yield non-rigid classes in $K_3$? The Cheeger-Chern-Simons invariants are rigid, but are there other suitably analytic invariants, maybe related to regulators, that could do the job?

Any help, comments and explanations are most welcome.


1 Answer 1


In answer to question 3 it is not possible to go from an element of the Bloch group to a hyperbolic manifold as there are different three manifolds with the same element in the Bloch group. That is different hyperbolic manifolds which are Scissor congruent. (See Quantum Content of Gluing Equations section 1.2 https://arxiv.org/pdf/1202.6268.pdf) I'm not sure if the image of the hyperbolic three manifolds generate the Bloch group but see the introduction of section 4 of Extended Bloch Group and Cheeger-Chern-Simons Class (https://projecteuclid.org/euclid.gt/1513883371) for a description of the additional data on closed three manifolds that would generate it or even better its lift.


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