, Volume 345, Issue 2, pp 453-489
Date: 08 Apr 2009

The topology of moduli spaces of free group representations

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For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character 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 \({{\rm SLm}(3, mathbb {C})}\) -character variety of a rank 2 free group is homotopic to an 8 sphere and the \({{\rm SLm}(2, mathbb {C})}\) -character variety of a rank 3 free group is homotopic to a 6 sphere.