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Incommensurability criteria for Kleinian groups

Author: James W. Anderson
Journal: Proc. Amer. Math. Soc. 130 (2002), 253-258
MSC (1991): Primary 57M50, 30F40; Secondary 20H10
Published electronically: April 26, 2001
MathSciNet review: 1855643
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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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Additional Information

James W. Anderson
Affiliation: Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, England

Keywords: Kleinian group, hyperbolic 3-manifold, commensurable
Received by editor(s): May 18, 2000
Published electronically: April 26, 2001
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2001 American Mathematical Society