DEFORMATION OF PROPERLY DISCONTINUOUS ACTIONS OF ℤ^k ON ℝ^k+1

Abstract

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a 'small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on 'stability' and 'local rigidity' of discontinuous groups. As a test case, we give an explicit description of the deformation space of ℤk acting properly discontinuously on ℝk+1 by affine nilpotent transformations. Our method uses an idea of 'continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

Deforming discontinuous subgroups for threadlike homogeneous spaces

“…Furthermore and unlike the context of Heisenberg groups, the topological stability fails to hold even generically on R( , G, H ). The present setup appears therefore as a new instance to ascertain these facts as shown in the papers [1,3,19]. On the other hand and according to the works [4,16,24,26,28,29], the proper action of a connected Lie subgroup L on nilpotent homogeneous spaces is equivalent to its free action whenever the nilpotent Lie group in question is of step strictly less than four.…”

“…The homomorphism ϕ is said to be topologically stable or merely stable in the sense of ), if there is an open set in Hom( , G) which contains ϕ and is contained in R( , G, H ). When the set R( , G, H ) is an open subset of Hom( , G), then obviously each of its elements is stable which is the case for any irreducible Riemannian symmetric spaces with the assumption that is torsion free uniform lattice of G [19,27]. Furthermore, we point out in this setting that the concept of stability may be one fundamental genesis to understand the local structure of the deformation space.…”

“…Kobayashi [17] generalized it in the case where H is not compact. For non-compact setting, the local rigidity does not hold in general in the non-Riemannian case and has been studied in [1,3,15,16,19]. We briefly recall here some details.…”

“…We briefly recall here some details. For a comprehensible information, we refer the readers to the references [1,3,7,10,[12][13][14][15][16][17][18][19] and some references therein. For ϕ ∈ R( , G, H ), the discontinuous subgroup ϕ( ) for the homogeneous space G/H is said to be locally rigid ( [17]) as a discontinuous group of G/H if the orbit of ϕ through the inner conjugation is open in R( , G, H ).…”

“…(See Problem C in [15] for further perspectives and basic examples). When it comes to the solvable context, our starting milestone is the paper [19] in which Kobayashi and Nasrin gave a concrete and explicit description of the deformation space of Z k acting properly discontinuously on G/H R k+1 by affine transformations, here G stands for a two-step nilpotent Lie group. The policy of these authors basically consists in building up an accurate cross-section of the adjoint orbits of elements of the parameter space R( , G, H ) of Hom( , G) which defines all the possible deformations of in such a way that the deformed subgroups preserve the property of being discontinuous subgroups for the homogeneous space G/H .…”