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Topology and singularities of free group character varieties

Monday 14th March 2011 - 16:30 to 17:30
We will discuss some generalities of the geometry, topology and singularities of the the G-character variety of F, that is, the moduli space Hom(F,G)/G of representations of a finitely presented group F into a Lie group G.

Then, we concentrate on the case when G is a complex affine reductive Lie group with maximal compact subgroup K, and F is a free group of rank r. In this situation, it can be proved that Hom(F,K)/K is a strong deformation retract of Hom(F,G)/G; in particular, both spaces have the same homotopy type. In the case G=SL(n,C), one can explicitly describe the singular locus of these character varieties, showing that they have the homotopy type of a manifold only when F or G are abelian, or r+n
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons