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Summary: Algebraic & Geometric Topology 11 (2011) 605624 605
Commensurators of
finitely generated nonfree Kleinian groups
CHRISTOPHER LEININGER
DARREN D LONG
ALAN W REID
We show that any finitely generated torsion-free nonfree Kleinian group of the
first kind which is not a lattice and contains no parabolic elements has discrete
commensurator.
20H10; 20F60, 57M50
1 Introduction
Let G be a group and 1; 2
if 1 \ 2 has finite index in both 1 and 2 . The commensurator of a subgroup
< G is defined to be
CG./ D f g 2 G W gg 1
is commensurable with g:
When G is a semisimple Lie group, and a lattice, a fundamental dichotomy es-
tablished by Margulis [26], determines that CG./ is dense in G if and only if is
arithmetic, and moreover, when is nonarithmetic, CG./ is again a lattice.
Historically, the prominence of the commensurator was due in large part to its density
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