scholarly journals Revisiting Poincaré’s Theorem on presentations of discontinuous groups via fundamental polyhedra

2015 ◽  
Vol 33 (4) ◽  
pp. 401-430
Author(s):  
E. Jespers ◽  
A. Kiefer ◽  
Á. del Río
Keyword(s):  
Discontinuous Groups

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

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Ali Baklouti ◽  
Sonia Ghaouar ◽  
Fatma Khlif
Keyword(s):  
Discontinuous Groups

AbstractLet ℍ

Planar discontinuous groups

1980 ◽  
pp. 106-177 ◽  
Author(s):  
Heiner Zieschang ◽  
Elmar Vogt ◽  
Hans-Dieter Coldewey
Keyword(s):  
Discontinuous Groups

scholarly journals Discontinuous groups of affine transformations of $C^{3}$

1976 ◽  
Vol 28 (1) ◽  
pp. 89-94 ◽  
Author(s):  
Kôji Uchida ◽  
Hisao Yoshihara
Keyword(s):  
Affine Transformations ◽  
Discontinuous Groups

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

2016 ◽  
Vol 27 (10) ◽  
pp. 1650085
Author(s):  
A. Baklouti ◽  
N. Elaloui ◽  
I. Kedim
Keyword(s):  
Homogeneous Space ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Rigidity Theorem ◽  
General Context ◽  
Affine Transformations ◽  
Discontinuous Group ◽  
Solvable Case ◽  
Local Rigidity ◽  
Discontinuous Groups

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.

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