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Abstract
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Using the wonderful compactification of a semisimple adjoint affine algebraic group
defined over an
algebraically closed field
of arbitrary characteristic, we construct a natural compactification
of the
-character variety of any
finitely generated group
.
When
is a free group, we show that this compactification is always simply
connected with respect to the étale fundamental group, and when
it is also topologically simply connected. For other groups
,
we describe conditions for the compactification of the moduli space to
be simply connected and give examples when these conditions are
satisfied, including closed surface groups and free abelian groups when
. Additionally,
when
is a free group we identify the boundary divisors of
in terms
of previously studied moduli spaces, and we construct a family of Poisson structures on
and its
boundary divisors arising from Belavin–Drinfeld splittings of the double of the Lie algebra
of
. In
the appendix, we explain how to put a Poisson structure on a quotient of a
Poisson algebraic variety by the action of a reductive Poisson algebraic
group.
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Keywords
character variety, wonderful compactification, moduli
space, fundamental group, Poisson
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Mathematical Subject Classification 2010
Primary: 14D20, 14F35, 14L30, 14M27, 53D17
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Milestones
Received: 29 October 2017
Revised: 1 February 2019
Accepted: 4 February 2019
Published: 27 November 2019
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