## Hiroshima Mathematical Journal

### Some Problems of deformations on three-step nilpotent Lie groups

#### Abstract

Let $G$ be an exponential solvable Lie group and $H$ a connected Lie subgroup of $G$. Given any discontinuous group $\mathit{\Gamma}$ for the homogeneous space $\mathscr M = G/H$ and any deformation of $\mathit{\Gamma}$, deformation of discrete subgroups may destroy proper discontinuity of the action on $\mathscr M$ as $H$ is not compact (except the case when it is trivial). To interpret this phenomenon in the case when $G$ is a 3-step nilpotent, we provide a layering of Kobayashi’s deformation space $\mathscr T(\mathit{\Gamma}, G, H)$ into Hausdorff spaces, which depends upon the dimensions of $G$-adjoint orbits of the corresponding parameter space. This allows us to establish a Hausdorffness theorem for $\mathscr T(\mathit{\Gamma}, G, H)$.

#### Article information

Source
Hiroshima Math. J., Volume 49, Number 2 (2019), 195-233.

Dates
Revised: 2 November 2018
First available in Project Euclid: 26 July 2019

https://projecteuclid.org/euclid.hmj/1564106545

Digital Object Identifier
doi:10.32917/hmj/1564106545

Mathematical Reviews number (MathSciNet)
MR3984992

Subjects
Primary: 12A34 98B76
Secondary: 23C57

#### Citation

Baklouti, Ali; Boussoffara, Mariem; Kedim, Imed. Some Problems of deformations on three-step nilpotent Lie groups. Hiroshima Math. J. 49 (2019), no. 2, 195--233. doi:10.32917/hmj/1564106545. https://projecteuclid.org/euclid.hmj/1564106545