Incommensurability criteria for Kleinian groups

Anderson, James

doi:10.1090/S0002-9939-01-06076-2

Incommensurability criteria for Kleinian groups
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by James W. Anderson PDF

Proc. Amer. Math. Soc. 130 (2002), 253-258 Request permission

Abstract:

The purpose of this note is to present a criterion for an infinite collection of distinct hyperbolic 3-manifolds to be commensurably infinite. (Here, a collection of hyperbolic 3-manifolds is commensurably infinite if it contains representatives from infinitely many commensurability classes.) Namely, such a collection $\mathbf {M}$ is commensurably infinite if there is a uniform upper bound on the volumes of the manifolds in $\mathbf {M}$. There is a related criterion for an infinite collection of distinct finitely generated Kleinian groups with non-empty domain of discontinuity to be commensurably infinite. (Here, a collection of Kleinian groups is commensurably infinite if it is infinite modulo the combined equivalence relations of commensurability and conjugacy in $\operatorname {Isom}^+(\mathbf {H}^3)$.) Namely, such a collection $\mathbf {G}$ is commensurably infinite if there is a uniform bound on the areas of the quotient surfaces of the groups in $\mathbf {G}$.

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