Three-dimensional Sol manifolds and complex Kleinian groups

Waldemar Barrera, Rene Garcia-Lara and Juan Navarrete

Vol. 294 (2018), No. 1, 1–18

DOI: 10.2140/pjm.2018.294.1

Abstract

We give a topological description of the quotient space $\Omega \left(G\right)∕G$, in the case when $G\subset \mathrm{\text{ PSL}}\left(3,ℂ\right)$ is a discrete subgroup acting on ${\mathbb{P}}_{ℂ}^2$ and the maximum number of complex projective lines in general position contained in the Kulkarni limit set $\Lambda \left(G\right)={\mathbb{P}}_{ℂ}^2\setminus \Omega \left(G\right)$ is equal to 4. Moreover, we give a topological description of the quotient space $\Omega \left(G\right)∕G$ in the case when $G$ is a lattice of the Heisenberg group.

Keywords

Kleinian groups, projective complex plane, discrete groups, limit set

Mathematical Subject Classification 2010

Primary: 32F45, 37F30, 32Q45

Secondary: 22E40, 37F45

Milestones

Received: 20 January 2017

Revised: 5 June 2017

Accepted: 26 September 2017

Published: 5 January 2018