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Bipolar Coxeter groups
We consider the class of those Coxeter groups for which removing from the
Cayley graph any tubular neighbourhood of any wall leaves exactly two connected
components. We call these Coxeter groups bipolar. They include both the
virtually Poincare duality Coxeter groups and the infinite irreducible
2-spherical ones. We show in a geometric way that a bipolar Coxeter group
admits a unique conjugacy class of Coxeter generating sets. Moreover, we
provide a characterisation of bipolar Coxeter groups in terms of the associated
Coxeter diagram.Comment: 25 pages, 2 figure
Quasi-isometric diversity of marked groups
We use basic tools of descriptive set theory to prove that a closed set
of marked groups has quasi-isometry classes
provided every non-empty open subset of contains at least two
non-quasi-isometric groups. It follows that every perfect set of marked groups
having a dense subset of finitely presented groups contains
quasi-isometry classes. These results account for most known constructions of
continuous families of non-quasi-isometric finitely generated groups. They can
also be used to prove the existence of quasi-isometry classes of
finitely generated groups having interesting algebraic, geometric, or
model-theoretic properties.Comment: Minor corrections. To appear in the Journal of Topolog
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