Narrow Results

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL₂ℂ inherits the structure of an algebraic variety known as the representation variety of G in SL₂ℂ. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between ismorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n if and only if Dim(R(G)) = 3n. In fact, a trivial consequence of Theorem 1.8 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2, there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation 1 with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups.

Let us consider the group G = 〈x, y | x^m = yⁿ〉 with m and n nonzero integers. The set R(G) of representations of G over SL(2, ℂ) is a four-dimensional algebraic variety which is an invariant of G. In this paper the number of irreducible components of R(G) together with their dimensions are computed. We also study the set of metabelian representations of this family of groups. Finally, the behavior of the projection t : R(G) → X(G), where X(G) is the character variety of the group, and some combinatorial aspects of R(G) are investigated.

If G is a finitely generated group and A is an algebraic group, then R_A(G) = Hom(G, A) is an algebraic variety. Define the "dimension sequence" of G over A as P_d(R_A(G)) = (N_d(R_A(G)), …, N₀(R_A(G))), where N_i(R_A(G)) is the number of irreducible components of R_A(G) of dimension i (0 ≤ i ≤ d) and d = Dim(R_A(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following:

Theorem A.Let w be a nontrivial word in the commutator subgroup ofF_n = 〈x₁, …, x_n〉, and letG = 〈x₁, …, x_n; w = 1〉. IfR_{SL(2, ℂ)}(G)is an irreducible variety andV_-1 = {ρ | ρ ∈ R_{SL(2, ℂ)}(F_n), ρ(w) = -I} ≠ ∅, thenP_d(R_{SL(2, ℂ)}(G)) ≠ P_d(R_{PSL(2, ℂ)}(G)).

Theorem B.Let w be a nontrivial word in the free group on{x₁, …, x_n}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x₁, …, x_n; w = 1〉. IfR_{SL(2, ℂ)}(G)is an irreducible variety, thenP_d(R_{SL(2, ℂ)}(G)) ≠ P_d(R_{PSL(2, ℂ)}(G)).

We also show that if G = 〈x₁, . ., x_n, y; W = y^p〉, where p ≥ 1 and W is a word in F_n = 〈x₁, …, x_n〉, and A = PSL(2, ℂ), then Dim(R_A(G)) = Max{3n, Dim(R_A(G′)) +2 } ≤ 3n + 1 for G′ = 〈x₁, …, x_n; W = 1〉.

Another one of our results is that if G is a torus knot group with presentation 〈x, y; x^p = y^t〉 then P_d(R_{SL(2, ℂ)}(G))≠P_d(R_{PSL(2, ℂ)}(G)).

Motivated by a possible connection between the SU(N) instanton knot Floer homology of Kronheimer and Mrowka and 𝔰𝔩(N) Khovanov–Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colorings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yamada in their state-sum interpretation of the quantum 𝔰𝔩(N) knot polynomial. For graphs with two colors, they showed this moduli space can be thought of as a representation variety, and that its Euler characteristic is equal to the 𝔰𝔩(N) polynomial of the graph evaluated at 1. We extend their results to graphs with arbitrary colorings by irreducible anti-symmetric representations of 𝔰𝔩(N).

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

Let $\mathrm{\Gamma }=〈a,b|a{b}^p{a}^{-1}=b^q〉$ be a Baumslag–Solitar group and let $G$ be a complex reductive algebraic group with maximal compact subgroup $K<G$. We show that, when $p$ and $q$ are relatively prime with distinct absolute values, there is a strong deformation retraction of $\mathrm{\text{Hom}}(\mathrm{\Gamma },G)$ onto $\mathrm{\text{Hom}}(\mathrm{\Gamma },K)$.

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable connections, and of holomorphic integrable connections over a compact Kähler manifold. In addition, we describe the moduli space of Higgs bundles and how it relates to the representation variety. We attempt to avoid abstraction, but strive to present and clarify the unifying ideas underlying the theory.