Abstract. For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the Gcharacter variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3, )-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2, )-character variety of a rank 3 free group is homotopic to a 6 sphere.

Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessary and sufficient conditions for the character variety Hom(A,G)//G to be irreducible when G is connected and semisimple. For a general connected reductive G, analogous conditions are found to be sufficient for irreducibility, when A is free abelian.Comment: 33 pages; version 3: few small changes, one error corrected, one or two additional references; to appear in Topology and its Application

Let X r be the moduli space of SL n , SU n , GL n , or U n -valued representations of a rank r free group. We classify the algebraic singular stratification of X r . This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating algebraic singularities to topological singularities, we show the moduli spaces X r generally are not topological manifolds, except for a few examples we explicitly describe.

Abstract. The SL(3, C)-representation variety R of a free group F r arises naturally by considering surface group representations for a surface with boundary. There is an SL(3, C)-action on the coordinate ring of R by conjugation. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. We show the coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F r imposed by the surface. Lastly, we work out the bracket on all generators when the surface is a three-holed sphere or a one-holed torus.

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