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

DOI:
https://doi.org/10.1090/S0002-9939-01-06076-2

Published electronically:
April 26, 2001

MathSciNet review:
1855643

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Abstract | References | Similar Articles | Additional Information

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 is commensurably infinite if there is a uniform upper bound on the volumes of the manifolds in .

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 .) Namely, such a collection is commensurably infinite if there is a uniform bound on the areas of the quotient surfaces of the groups in .

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Additional Information

**James W. Anderson**

Affiliation:
Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, England

Email:
j.w.anderson@maths.soton.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-01-06076-2

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