Representations of fundamental groups of surface


Let S be a closed oriented surface of genus g (“a g-holed torus”). The fundamental group of S gas 2g generators a₁, b₁, . . . , ag, bg subject to the single relation

a₁b₁a₁-1b₁-1 …agbgag-1bg-1 = 1. (1)

A representation of the fundamental group of S in SL(2,R) is given by a choice of 2g real 2 × 2-matrices with determinant 1, satisfying the relation (1). To each such representation one can associate an integer invariant called the Toledo invariant. A theorem of J. Milnor says that the absolute value of the Toledo invariant is less than or equal to g − 1, and a theorem of W. Goldman says that two representations can be continuously deformed into each other if and only if they have the same Toledo invariant.

In the seminar we shall explain the concepts and results mentioned above. Moreover, time permitting, we shal indicate how methods of holomorphic geometry can be used to study such questions through so-called Higgs bundles.