Groups and geometry

The representations of the fundamental group of an algebraic variety form important topological invariants connecting group theory with geometry. In this talk, we focus on the varieties of smallest non-trivial dimension: curves. Using quivers and jet schemes, we show that the geometry of the space of representations has constrained singularities. We apply this to show that the number of irreducible complex representations of SL_n(Z) of dimension at most m grows at most as the square of m, for a fixed n>2.