## Kyoto Journal of Mathematics

### Deforming discontinuous subgroups of reduced Heisenberg groups

#### Abstract

Let $G=\mathbb{H}^{r}_{2n+1}$ be the $(2n+1)$-dimensional reduced Heisenberg group, and let $H$ be an arbitrary connected Lie subgroup of $G$. Given any discontinuous subgroup $\Gamma\subset G$ for $G/H$, we show that resulting deformation space $\mathscr{T}(\Gamma,G,H)$ of the natural action of $\Gamma$ on $G/H$ is endowed with a smooth manifold structure and is a disjoint union of open smooth manifolds. Unlike the setting of simply connected Heisenberg groups, we show that the stability property holds and that any discrete subgroup of $G$ is stable, following the notion of stability. On the other hand, a local (and hence global) rigidity theorem is obtained. That is, the related parameter space $\mathscr{R}(\Gamma,G,H)$ admits a rigid point if and only if $\Gamma$ is finite.

#### Article information

Source
Kyoto J. Math., Volume 55, Number 1 (2015), 219-242.

Dates
First available in Project Euclid: 13 March 2015

https://projecteuclid.org/euclid.kjm/1426252136

Digital Object Identifier
doi:10.1215/21562261-2848169

Mathematical Reviews number (MathSciNet)
MR3323533

Zentralblatt MATH identifier
1317.22006

#### Citation

Baklouti, Ali; Ghaouar, Sonia; Khlif, Fatma. Deforming discontinuous subgroups of reduced Heisenberg groups. Kyoto J. Math. 55 (2015), no. 1, 219--242. doi:10.1215/21562261-2848169. https://projecteuclid.org/euclid.kjm/1426252136

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