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Abstract:
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(Joint with Dani Wise)
If a group G acts on
a CAT(0) cube complex, the action induces a very
rich structure on
the group. Sageev showed that actions on CAT(0) cube
complexes correspond
to "codimension-1" subgroups, which coarsely
separate the group
into two complementary components. We think of
these components as
"halfspaces" and the cutting subgroup as a "wall".
Some groups admit
codimension-1 subgroups, and some don't. Indeed the
existence of actions
on cube complexes is closely related to
representation theoretic
notions such as Kazhdan's Property (T) and
its strong negation,
a-T-menability.
I will discuss
"cubulations" of groups: how to construct them and how
to recognize some of
their basic finiteness properties.
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