WHEN IS THE DEFORMATION SPACE $\mathscr{T}(\Gamma, H_{2n+1}, H)$ A SMOOTH MANIFOLD?

Abstract

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space [Formula: see text] of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then [Formula: see text] splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that [Formula: see text] admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

Open Problems in Deformation Theory of Discontinuous Groups Acting on Homogeneous Spaces

“…Theorem 3.4 (cf. [5]). Let H = exp(h) be a connected subgroup of the Heisenberg group G = exp(g) and Γ a discontinuous subgroup of G for the homogeneous space G/H with a syndetic hull L = exp(l).…”

“…In the setting of Heisenberg groups, we got in [5] an answer to question 5.3. We proved the following: Theorem 5.6 (cf.…”

“…We proved the following: Theorem 5.6 (cf. [5]). Let Γ be a discontinuous subgroup of the Heisenberg group G = exp(g) and exp(l) its syndetic hull.…”

“…The problem of deformation consists in seeking how to deform Γ by means of homomorphisms from Γ to G (thus to consider the set Hom(Γ, G) of all these homomorphisms) in a way such that the deformed discrete subgroup acts properly on G/H. The problem of describing deformations was first advocated by T. Kobayashi in [21] for the general non-Riemannian setting and precisely determines as proposed in [5], the set of deformation parameters that allow Γ to deform in a way to guarantee the proper discontinuity on G/H. (endowed with the point wise convergence topology), rather than Hom(Γ, G), plays a crucial role in these problems.…”

“…The analytic variety Hom(Γ, G) may have some singularities and there is no clear raison, to say that the parameter space R(Γ, G, H) is an analytic or algebraic or smooth manifold. It has been shown in [5] and [4] that the set Hom 0 (Γ, G)/G, (where Hom 0 (Γ, G) designates the set of injective homomorphisms) is endowed with a smooth manifold structure in some nilpotent contexts. In general, the action of G on the space Hom(Γ, G) is not proper, then the quotient spaces Hom(Γ, G)/G and T (Γ, G, H) may fail to be Hausdorff spaces.…”