Counting commensurability classes of hyperbolic manifolds

Subgroup growth usually means the asymptotic behavior of the number of subgroups of index n of a given f.g. group as a function of n.

We generalize this to discrete (torsion-free) subgroups of the Lie group G=SO+(n,1) for which the quotient admits finite volume, as a function of the co-volume. Conjugacy classes of such discrete subgroups correspond geometrically to n-dimensional hyperbolic manifolds of finite volume.

By a classical result of Wang, for n >=4 there are only finitely many such conjugacy classes up to any given finite volume V. More recently, Burger, Gelander, Lubotzky and Mozes showed that this number grows like V^V.

 

In this talk we focus on counting commensurability classes. Two subgroups are commensurable if they admit a common finite index subgroup (in our context, up to taking conjugates). We show that surprisingly, for n >= 4 this number grows like V^V as well. Since the number of arithmetic commensurability classes grows ~polynomially (Belolipetsky), our result implies that non-arithmetic subgroups account for “most" commensurability classes.

 

Our proof uses a mixture of arithmetic, hyperbolic geometry and some combinatorics. In particular, recall that a quadratic form of signature (n,1) over a totally real number field, whose conjugates are positive definite, defines an arithmetic discrete subgroup of finite covolume in G. As in the classical construction of Gromov--Piatetski-Shapiro, several non-similar quadratic forms can be combined to construct amalgamated non-arithmetic subgroups.