DEFORMATIONS OF MAXIMAL REPRESENTATIONS IN Sp(4,R)

A good way to understand an object of study is, as Richard Feynman famously remarked, to “just look at the thing!”. In this paper we apply Feynman’s method to answer the following question: given a surface group representation in Sp(4,R), under what conditions can it be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)? A surface group representation in a groupG is a homomorphism from the fundamental group of the surface into G. For a surface of genus g > 2, the moduli space of reductive surface group representations into G = Sp(4,R), denoted by R(Sp(4,R)), has 3 ·22g+1+8g−13 connected components (see [16, 21]). The components are partially labeled by an integer, known as the Toledo invariant, which ranges between 2 − 2g and 2g−2. If Rd denotes the component with Toledo invariant d, then there is a homeomorphism Rd ≃ R−d and except for the extremal cases (i.e. |d| = 2g − 2) each Rd is connected. In contrast, the subspaces of maximal representations Rmax = R±(2g−2) have 3 · 2 + 2g − 4