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.
Keywords
character variety, wonderful compactification, moduli
space, fundamental group, Poisson