Teardrop And Sourcil Line (TSL): A Novel Radiographic Sign That Predicts Residual Acetabular Dysplasia in DDH After Closed Reduction

Background: Residual acetabular dysplasia (RAD) is a major problem of developmental dysplasia of the hip (DDH) after closed reduction (CR). Several parameters have been investigated as ways of predicting RAD; however, early prediction of RAD remains controversial. The purpose of this study was to evaluate the radiographic sign of teardrop and sourcil line (TSL) in pediatric patients with DDH to enable prediction of RAD after CR early.Methods: One hundred and twenty-five hips with DDH treated with CR and followed up for at least 2 years were included in this study. The mean age at CR was 18.3 months (range, 9 to 32 months) and the average follow-up time was 44.2 months (range, 24 to 83 months). The acetabular index (AI) was measured at different time points. RAD was determined according to the modified Severin criteria. The cases were divided into two groups according to whether TSL became continuous or not. The relationships among TSL, AI and RAD were analyzed.Results: The RAD incidence was 73.6% (92/125) at the last follow-up. AI at CR and TSL were the prognostic factors for RAD (p=0.017 and 0.001, respectively). Thirty-four hips showed a continuous TSL. The mean time when TSL became continuous after CR was 20.9 months (range, 8 to 57 months). There was a lower RAD rate in the TSL continuous group (p<0.001). There was no statistical difference in the AI at CR between the TSL continuous and discontinuous groups; however, the level of AI after CR was lower in the TSL continuous group. In the TSL continuous group, there was no significant difference in the time at which TSL became continuous after CR between RAD and non-RAD hips.Conclusions: The TSL continuous group had a lower AI and incidence of RAD than the discontinuous group. The TSL can be a predictive factor of RAD in DDH after CR and can predict RAD at an earlier time than AI measurement.

Deforming discontinuous groups for Heisenberg motion groups

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group [Formula: see text] acting on a homogeneous space [Formula: see text], where [Formula: see text] stands for a closed subgroup of the Heisenberg motion group [Formula: see text]. That is, the parameter space admits a locally rigid (equivalently a strongly locally rigid) point if and only if [Formula: see text] is finite. Moreover, Calabi–Markus’s phenomenon and the question of existence of compact Clifford–Klein forms are also studied.

Determination of the Homology and the Cohomology of a Few Groups of Isometries of the Hyperbolic Plane

In this paper we determine the homology and the cohomology groups of two properly discontinuous groups of isometries of the hyperbolic plane having non-compact orbit spaces and the fundamental group of a graph of groups with a finite vertex groups and no trivial edges by extending Lyndon’s partial free resolution for finitely presented groups. For the first two groups, we obtain partial extensions and the corresponding homology. We also compute the corresponding cohomology groups for one of these groups. Finally we obtain homology and cohomology in all dimensions for the last of the above mentioned groups by constructing a full resolution for this group.GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 65-77

Variants of stability of discontinuous groups for Euclidean motion groups

Let [Formula: see text] be a Lie group, [Formula: see text] a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. Given a deformation parameter [Formula: see text], the deformed subgroup [Formula: see text] may fail to act properly discontinuously on [Formula: see text]. To understand this phenomenon in the case when [Formula: see text] stands for an Euclidean motion group [Formula: see text], we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of [Formula: see text] on [Formula: see text], Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when [Formula: see text] turns out to be a crystallographic subgroup of [Formula: see text].

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Homogeneous space with non-virtually abelian discontinuous groups but without any proper SL(2, ℝ)-action

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space [Formula: see text] admits a discontinuous group which is not virtually abelian if and only if [Formula: see text] admits a proper [Formula: see text]-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces [Formula: see text] of reductive type. We give a counterexample in the case [Formula: see text].

Stability of discontinuous groups for reduced threadlike Lie groups

Let G be a reduced threadlike Lie group, H an arbitrary closed connected subgroup of G and Γ ⊂ G an abelian discontinuous subgroup for G/H. We study in this work some topological properties of the parameter space [Formula: see text] and the deformation space [Formula: see text], namely the stability and the rigidity. Instead of treating stability, we consider a weaker form by using an explicit covering of Hom (Γ, G) which we call layering and we show that the local rigidity holds if and only if Γ is finite.

On discontinuous groups acting on (ℍ2n+1r × ℍ2n+1r)/Δ

Let ℍ