Rank two jump loci for solvmanifolds and Lie algebras

Abstract

We consider representation varieties in SL 2 for lattices in solvable Lie groups, and representation varieties in sl 2 for finite-dimensional Lie algebras. Inside them, we examine depth 1 characteristic varieties for solvmanifolds, respectively resonance varieties for cochain Differential Graded Algebras of Lie algebras. We prove a general result that leads, in both cases, to the complete description of the analytic germs at the origin, for the corresponding embedded rank 2 jump loci.

Infinitesimal finiteness obstructions

“…Likewise, suppose that H < π is a finite-index subgroup. Then, as noted in [26,Lemma 3.6], the inclusion α : H → π induces a morphismα : π → H with finite kernel, which sends V i r (π) to V i r (H) for all i, r 0. For the free groups F n of rank n 2, we have that V 1 r (F n ) = (C × ) n for r n − 1 and V 1 n (F n ) = {1}.…”

“…Likewise, suppose that is a finite‐index subgroup. Then, as noted in [, Lemma 3.6], the inclusion induces a morphism with finite kernel, which sends to for all .…”

“…Proof of Theorem 5.4. Once again, let f : p C phpAqq Ñ A be the cdga map from (26). We need to show that H 1 p f q is an isomorphism and H 2 p f q is a monomorphism.…”

Around the Tangent Cone Theorem

“…Let Γ be a discrete, co-compact subgroup of a simply-connected, solvable, real Lie group G, and let M = G/Γ be the corresponding solvmanifold. As shown in [37,53], all the characteristic varieties of M are finite subsets of Char(M ). Moreover, as shown by Papadima and Pȃunescu in [53], if (A, d) is any finite-dimensional model for M (such as the one constructed by Kasuya [37]), then all the resonance varieties R i (A) contain 0 as an isolated point; in particular, TC 1 (V i (M )) = TC 0 (R i (A)) = {0}.…”

“…Now suppose G is completely solvable, and (A, d) is the classical Hattori model for the solvmanifold M = G/Γ. Work of Millionschikov [49], as reprised in [53], shows that R i (A) is also a finite set. Furthermore, there are examples of solvmanifolds of this type where R i (A) is different from {0}.…”