Title: On representation varieties of 3–manifold groups
Abstract: We prove universality theorems ("Murphy's laws") for representation varieties of fundamental groups of closed 3-dimensional manifolds.We show that germs of SL.2/representation schemes of such groups are essentially the same as germs of schemes over Q of finite type.14B12, 20F29, 57M05 IntroductionIn this paper we will prove that there are no restrictions on local geometry of representation schemes of 3-manifold groups to PO.3/ and SL.2/.Note that both groups H D PO.3/ and H D SL.2/ are affine algebraic group schemes defined over Q; thus, for every finitely generated group , the representation schemes Hom.; H / and character schemes X.; H / D Hom.; H / = = H are affine algebraic schemes over Q.Our goal is to show that, to some extent, these are the only restrictions on local geometry of the representation and character schemes of fundamental groups of closed 3-manifolds.The universality theorem we thus obtain is one of many universality theorems about moduli spaces of geometric objects; see Mnëv [11], Richter-Gebert [15], Kapovich and Millson [6; 7; 8], Vakil [18], Payne [13], Rapinchuk [14].Below is the precise formulation of our universality theorem.In what follows we use the notation G D PO.3/ and z G D Spin.3/.Theorem 1.1 Let X C N be an affine algebraic scheme over Q and let x 2 X be a rational point.Then there exist1. an open subscheme X 0 X containing x , 2. a closed 3-dimensional manifold M with fundamental group ,