Non-Virtually Abelian Discontinuous Group Actions vs. Proper SL(2,ℝ) -Actions on Homogeneous Spaces

Abstract

We develop algorithms and computer programs which verify criteria of properness of discrete group actions on semisimple homogeneous spaces. We apply these algorithms to find new examples of non-virtually abelian discontinuous group actions on homogeneous spaces which do not admit proper $\mathit{\text{SL}}(2,\mathbb{R})$-actions.

Keywords:

• proper action
• homogeneous space
• semisimple Lie algebra
• GAP4
• computing with simple Lie algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B05
• 17B10
• 53C30
• 57S25
• 57S30

Acknowledgments

We cordially thank the referees for valuable advice.

Additional information

Funding

The first named author was supported by the National Science Center, Poland, grant NCN, no. 2018/31/D/ST1/00083. The fourth named author was supported by the National Science Center, Poland, grant no. 2018/31/B/ST1/00053.