Maximal Representations, real Spectrum, and harmonic Maps

The character variety of maximal Sp(2n,R)-representations of the fundamental group G of a surface S is homeomorphic to a real semi algebraic set X and as such admits a compactification by the set of closed points in the real spectrum of the coordinate ring of X. This compactification has very good topological properties and the mapping class group acts on it with virtually abelian stabilizers. Its ideal points are represented by certain Sp(2n,F) representations of G over various real closed fields F. Characterizing these representations and understanding their geometric properties is an ongoing theme of our research. In this talk we focus on the R-building B(n,F) associated to Sp(2n,F) and will explain why the corresponding G-action is proper in the sense of Korevaar and Schoen, implying the existence of an equivariant harmonic map for every complex structure h on S. We also explain that if the systole of the G-action on B(n,F) is strictly positive, the energy functional, as a function of h, is proper on Teichmueller space. Such actions with strictly positive systole form an open set in the real spectrum boundary, that is non-empty as soon as n is at least 2. joint work with A.Iozzi, A.Parreau and B. Pozzetti.

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