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Coxeter group in Hilbert geometry
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A theorem of Tits and Vinberg allows to build an action of a Coxeter group
\Gamma
on a properly convex open set
\Omega
of the real projective space, thanks to the data
P
of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe a hypothesis that makes those conditions necessary.
Under this hypothesis, we describe the Zariski closure of
\Gamma
, nd the maximal
\Gamma
-invariant convex set, when there is a unique
\Gamma
-invariant convex set, when the convex set
\Omega
is strictly convex, when we can find a
\Gamma
-invariant convex set
\Omega
' which is strictly convex.
European Mathematical Society - EMS - Publishing House GmbH
Title: Coxeter group in Hilbert geometry
Description:
A theorem of Tits and Vinberg allows to build an action of a Coxeter group
\Gamma
on a properly convex open set
\Omega
of the real projective space, thanks to the data
P
of a polytope and reflection across its facets.
We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite.
We describe a hypothesis that makes those conditions necessary.
Under this hypothesis, we describe the Zariski closure of
\Gamma
, nd the maximal
\Gamma
-invariant convex set, when there is a unique
\Gamma
-invariant convex set, when the convex set
\Omega
is strictly convex, when we can find a
\Gamma
-invariant convex set
\Omega
' which is strictly convex.
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