It is not hard to check that as time increases, the modulus (42) moves along a semicircle in the upper half of the complex plane, with a center on the real axis. Such a curve is a geodesic in the natural Weil-Petersson (or Poincaré) metric on the torus moduli space [159, 135] . Because of the invariance under the mapping class group (50), however, the true physical motion in the moduli space of the torus - the space of physical configurations with the large diffeomorphisms modded out - is much more complicated; there are arbitrarily long geodesics, and the flow is, in fact, ergodic  .
For spacetimes with being a surface of genus , no explicit metrics analogous to Equation (40) are known, except for the special case of solutions with constant moduli. The problem is in part that no simple form such as Equation (41) for the “standard” constant curvature metrics exists, and in part that the ADM Hamiltonian becomes a complicated, nonlocal function of the moduli. For the case of an asymptotically flat genus space, some interesting progress has been made by Krasnov  ; I do not know whether these methods can be extended to the spatially closed case.
One can write down the holonomies of the geometric structure for a higher genus surface, of course - though even there, it is nontrivial to ensure that they represent spacetimes with spacelike genus slices - but to a physicist, these holonomies in themselves give fairly little insight into the dynamics. In principle, the ADM and Chern-Simons approaches might be viewed as complimentary: As Moncrief has pointed out, one could evaluate the holonomies in terms of ADM variables in a nice time-slicing, set these equal to constants, and thereby solve the ADM equations of motion  . In practice, though, this approach seems intractable except for the genus one case. For , it may be possible to extract a useful physical picture from the geometrical results of , which relate holonomies to the structure of the initial singularity and the asymptotic future geometry, but the implications have not yet been explored in any depth.
A number of qualitative statements nevertheless remain possible. The singular behavior of the torus universe carries over to higher genus: Spacetimes with expand from a big bang and recollapse in a big crunch, while those with expand forever [200, 20] . Moreover, the degeneration of the spatial geometry at the initial singularity carries over to the higher genus case [200, 55] . By introducing a global “cosmological time” and exploiting recent results in two- and three-dimensional geometry, Benedetti and Guadagnini have shown that when , a set of parameters describing the initial singularity and a second set describing the geometry in the asymptotic future together completely determine the spacetime  . It seems likely that these two sets are canonically conjugate, and a better understanding of the symplectic structure could be useful for quantum gravity.
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