Flawed groups and the topology of character varieties

Abstract

A finitely presented group Γ is called flawed if Hom(Γ, G)/ /G deformation retracts onto its subspace Hom(Γ, K)/K for reductive affine algebraic groups G and maximal compact subgroups K ⊂ G. After discussing generalities concerning flawed groups, we show that all finitely generated groups isomorphic to a free product of nilpotent groups are flawed. This unifies and generalizes all previously known classes of flawed groups. We also provide further evidence for the authors' conjecture that RAAGs (with torsion) are flawed. Lastly, we show direct products between finite groups and some flawed group are also flawed. These latter two theorems enlarge the known class of flawed groups.

Geometry of $\mathrm{SU}(3)$-character varieties of torus knots

“…where Y π = Ỹπ ∩ Y r are those representations from (5) with t,l det(ρ tl ) = 1. However, observe that in this setting we no longer have an analogous decomposition (6).…”

“…The properties of these subvarieties have been widely studied in the literature, as in [1] or [3]. Furthermore, in [6] (see also [2,4,5]) the authors proved that, when Γ is a free product of nilpotent groups or a star-shaped RAAG (Right Angled Artin Groups), the inclusion X(Γ, U(r)) → X(Γ, SL(r, C)) is a deformation retract for any reductive group G (a property called flawed). On the contrary, for Γ = π 1 (Σ g ), the fundamental group of a compact orientable surface Σ g of genus g ≥ 2, this inclusion is never a homotopy equivalence when G is reductive and non-abelian (it is said that Γ is a flawless group in the language of [6]).…”

“…Furthermore, in [6] (see also [2,4,5]) the authors proved that, when Γ is a free product of nilpotent groups or a star-shaped RAAG (Right Angled Artin Groups), the inclusion X(Γ, U(r)) → X(Γ, SL(r, C)) is a deformation retract for any reductive group G (a property called flawed). On the contrary, for Γ = π 1 (Σ g ), the fundamental group of a compact orientable surface Σ g of genus g ≥ 2, this inclusion is never a homotopy equivalence when G is reductive and non-abelian (it is said that Γ is a flawless group in the language of [6]). These character varieties have been extensively studied, as in [10] for g = 2 and G = SU (2) or in [7] regarding the ergodic properties of the action of the mapping class group of Σ g .…”

Mixed Hodge structures on character varieties of nilpotent groups

“…For an example where they disagree, let Γ = Γ ∠ be the "angle RAAG" associated with a path graph with 3 vertices, considered in [FL4]. Then, even for a low dimensional group such as G = SL(2, C) we have that M 0 Γ G has 3 irreducible components, one being M T Γ G and 2 extra ones.…”

“…Moreover, for the case G = SL(3, C) there are components in M 0 Γ G which have higher dimension than the dimension of M T Γ G. Remark 6. One can also ask if the identity representation is contained in a single irreducible component of M 0 Γ G. This also fails for M 0 Γ ∠ (SL(2, C)), as shown in [FL4]. 3.2.…”