Vol. 242, No. 1, 2009

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ISSN: 0030-8730
Obtaining the one-holed torus from pants: Duality in an SL(3, )-character variety

Sean Lawton

Vol. 242 (2009), No. 1, 131–142

The SL(3, )-representation variety R of a free group Fr arises naturally by considering surface group representations for a surface with boundary. There is an SL(3, )-action on the coordinate ring of R. 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. The coordinate ring [X] is a complex Poisson algebra with respect to a presentation of Fr imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map that turns the three-holed sphere into the one-holed torus induces a rank-preserving Poisson map on [X].

Poisson, character variety, free group
Mathematical Subject Classification 2000
Primary: 14L24
Secondary: 53D30
Received: 4 September 2008
Accepted: 12 May 2009
Published: 1 September 2009
Sean Lawton
Departamento de Matemática
Instituto Superior Técnico
Av. Rovisco Pais
1049-001 Lisboa