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Summary: Commensurators of finitely generated non-free Kleinian groups
C. J. Leininger, D. D. Long & A. W. Reid
September 2, 2009
1 Introduction
Let G be a group and 1, 2 < G. 1 and 2 are called commensurable if 1 2 has finite index
in both 1 and 2. The Commensurator of a subgroup < G is defined to be:
CG() = {g G : gg-1
is commensurable with }.
When G is a semi-simple Lie group, and a lattice, a fundamental dichotomy established by
Margulis [25], determines that CG() is dense in G if and only if is arithmetic, and moreover,
when is non-arithmetic, CG() is again a lattice.
Historically, the prominence of the commensurator, was due in large part to the density of the
commensurator in the arithmetic setting being closely related to the abundance of Hecke operators
attached to arithmetic lattices. These operators are fundamental objects in the theory of automor-
phic forms associated to arithmetic lattices (see [38] for example). More recently, the commensurator
of various classes of groups has come to the fore due its growing role in geometry, topology and ge-
ometric group theory; for example in classifying lattices up to quasi-isometry, classifying graph
manifolds up to quasi-isometry, and understanding Riemannian metrics admitting many "hidden
symmetries" (for more on these and other topics see [2], [4], [17], [18], [24], [34] and [37]).
In this article, we will study CG() when G = PSL(2, C) and a finitely generated non-
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