Counting Problems in Moduli Space

Recall that Ωn = SL(n,R)/SL(n,Z) is the space of covolume 1 lattices in R. This space is non-compact, since we can have arbitrarily short vectors in a lattice. We will refer to moduli spaces of translation surfaces as defined in the lectures by Howard Masur in this volume [Ma1, Definition 6] as strata. Note that the case of n = 2 in the space of lattices and the case of the stratum H1(∅) boil down to the same thing, since we are considering the space of unit area holomoprphic 1-forms on tori, which is given by SL(2,R)/SL(2,Z). Let B(R) be the ball of radius R centered at 0 in R. For a given lattice ∆ ∈ Ωn. we would like to find out how many lattice points, that is, how many points of ∆ are contained in B(R). It is immediately clear that for a fixed lattice ∆, as R → ∞,