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Hello all

As is probably well-known to most, in the upper halfplane we have a natural action of $SL_2(\mathbb{R})$ through linear fractional transformations, and a $2$-form $\frac{dx dy}{y^2}$ which is invariant under this action.

In the case of the upper half-space $H_3 = \{ (z,t) \ | \ z\in \mathbb{C} \ , \ t>0 \ \}$ we have an analogous situation, only now we have a natural action of all of $SL_2(\mathbb{C})$. The fastest way of defining this action is by identifying a point $(z,t)$ in $H_3$ with the quaternion $q:= z+t j$, and then setting $$ \begin{pmatrix} a&b\cr c& d\cr \end{pmatrix}.(z,t) := (aq+b)(cq+d)^{-1}$$ Now for my question; I want to try to construct an $SL_2(\mathbb{Z}[i])$-invariant two-form on $H_3$ which when restricted to the plane $Im(z)=0$ takes the form $\frac{dxdt}{t^2}$. Do such a thing exist ? How would I go about constructing it? Any thoughts/references would be greatly appreciated.

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  • $\begingroup$ You might be able to do this on a congruence subgroup of $PSL(2,\mathbb{Z}[i])$. Here's a reference: ams.org/mathscinet-getitem?mr=1123369 $\endgroup$
    – Ian Agol
    Commented Jan 17, 2011 at 21:07
  • $\begingroup$ Sorry, in my previous comment I assumed you were asking for an automorphic 2-form (like a harmonic form). If you just want a 2-form, you can definitely do this on a subgroup in which the totally geodesic surface becomes embedded and non-separating, such as the Whitehead or Borromean rings link complements. But it might be tricky to write down an explicit formula. $\endgroup$
    – Ian Agol
    Commented Jan 17, 2011 at 21:15

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In upper half space, the thing that's analogous to the invariant 2-form (which measures hyperbolic area) is the hyperbolic volume form $1/z^3 dx dy dz$.

If you want to understand 2-forms invariant by a particular discrete group, they are the same thing as 2-forms on the quotient orbifold. In the case of $SL_2$ of the algebraic integers in a quadratic imaginary field such as this, the groups are usually called Bianchi groups: Bianchi worked out fundamental domains for them for the first set of examples, as well as a general technique to find fundamental domains. On Allen Hatcher's website, there's a handy list of pictures of the folded-up fundamental domains in cases where they can be readily drawn.

In this case, the quotient orbifold has underlying space $S^3$ minus a point (for the cusp), with singular locus in the form of the 1-skeleton of a tetrahedron. Since the underlying space is contractible, any closed 2-form is a coboundary --- i.e. there are plenty of 2-forms, closed or otherwise, but there's no obvious reason they should be interesting. In this particular case, the diagonal matrix with entries $i, -i$ rotates 180 degrees about the $z$-axis, acting as an orientation-reversing map on the hyperbolic plane, so it takes the 2-form above to its negative: it's not invariant even before attempting to extend it.

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