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Main idea: According to Thurston, a "generic" conjugacy class R in T_g is pseudo-Anosov and have a canonical representative which preserves two tranversal laminations and multiplies the transversal measures by lambda, 1/lambda, where lambda (exponent of entropy) is strictly >1 and an algebraic integer (appears as the largest eignevalue on 1st cohomology of the ramified double covering on which the qudartic differential becomes an abelian differential). Moreover, one may assume that the laminations have 4g-4 triple singular points. I would like to construct a representation in which R acts non-trivially, because there will be some eigenvalue of the corresponding matrix equal to a positive power of lambda.
A representation of T_g is the same as a local system on M_g. Here is the description of it. The fiber of it at a complex curve C is a kind of middle (or total?) cohomology of the configuration space Conf_{4g-4}(C)=C^{4g-4}-Diagonals (the configuration space of 4g-4 points (x_1,...,x_{4g-4}) on C), with coefficients in the flat bundle, the 1st cohomology group of the double cover of C, doubly ramified at (x_1,...,x_{4g-4}).
Why I beleive that this should work: Thurston's representative gives a map from Conf_{4g-4}(C) (together with the local system) to itself, with a prefered marked point such that the action on the fiber has the largest eignevalue equal to lambda.
Can one use Lefschez fixed point formula, or maybe some dynamical reasoning, to see that this eigenvalue appear in the total cohomology of Conf_{4g-4}(C) with local coefficients as well? (MK)
Deformation quantization gives in general only an algebar over formal power series in Planck constant. Nevertheless, for a flat symplectic torus T the Moyal product in the integral form gives an actual family of associative products on smooth functions on T, depending on a real parameter hbar. On the other side, it is well-known that it is not possible to quantize the round 2-dimensional sphere, if one asks for an SO(3)-invariant product.
The question: can one actually quantize surface of negative curvature?
A possible candidate: F.Berezin quantized equivariantly the hyperbolic plane, using Toeplitz operators. Can one make sense for the star-product of Gamma-invariant functions where Gamma is a cocompact subgroup of PSL(2,R)?
It seems that some old calculaton of F.Radulescu confirms the convergence. If the answer²is yes, two more questions:
1) Will PSL(2,Z) transformations on hbar induce Morita equivalences?
2) for a given hbar, does there exists a flat connection on the corresponding bundle of algebras over the moduli space of complex sructures (=cocompact subgroups in PSL(2,R)? I expect a positive answer, it will give a new family of infinite-dimensional projective representations of the mapping class group T_g. (MK)
Let P be a rational polyhedron in R^n, contained in the standard unit cube. Fix also some integers (a_1,...,a_n) and an integral quadratic form Q in n variables. Such a gadget produces an sequence of algebraic numbers Z(k), in fact elements of cyclotomic fields.
Definition: for a positive integer k let q be a primitive k-th root of 1 (e.g; exp(2 Pi i/k)), for m in {0,...,k-1} denote by (m]! the q-factorial of m, =(1-q)(1-q^2)...(1-q^m).
Z(k):= sum over integer points m=(m_1,...,m_n) in kP of the product Prod_{i=1}^n ([m_i]!)^(a_i) times q^Q(m,m)
(Better notation would be Z(q), not Z(k)).
Let us consider a finite linear combinatioin with integer coefficients of such things. Obviusly, any quantum invariant of a 3-manifold is (as function of level) of such type.
I claim that ANY function as above has an asymptotic behavior as k--> infinity of the same nature as quantulm invariant. I mean that it will have an asymptotic expansion with the exponent equal to the Beilinson-Borel regulator of an element of K_3 (number field), times a series in 1/k with algebraic coefficients.
It leads to a bit non-precise problem: find linear relations between functions Z(q) arising from different polytopes P, vectors (a_1,..;,a_n) and quadratic forms Q.
Is there some universal group U, like scissor group such that an invariant of 3-manifold and a compact Lie group G is an element of U, and actual numerical valuesn of quantum invariants at different levels appear as its images of certain homomorphisms from U to cyclotomic fields? (MK)
Revised 19.12.2006
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