Kyoto Journal of Mathematics

Deforming discontinuous subgroups of reduced Heisenberg groups

Ali Baklouti, Sonia Ghaouar, and Fatma Khlif

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Let G=H2n+1r be the (2n+1)-dimensional reduced Heisenberg group, and let H be an arbitrary connected Lie subgroup of G. Given any discontinuous subgroup ΓG for G/H, we show that resulting deformation space T(Γ,G,H) of the natural action of Γ 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 R(Γ,G,H) admits a rigid point if and only if Γ is finite.

Article information

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

First available in Project Euclid: 13 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Secondary: 32G05: Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15]

Reduced Heisenberg group proper action free action deformation space rigidity


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.

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