Non-virtually abelian discontinuous group actions vs. proper SL(2,R)-actions on homogeneous spaces

TL;DR

This paper introduces algorithms to verify properness of discrete group actions on homogeneous spaces, leading to new examples of non-virtually abelian discontinuous actions that lack proper SL(2,R)-actions.

Contribution

It develops computational methods to identify non-virtually abelian discontinuous group actions and demonstrates their application with new examples.

Findings

Identified new non-virtually abelian discontinuous actions

Showed these actions do not admit proper SL(2,R)-actions

Provided algorithms for verifying properness of group actions

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 SL(2,R)-actions.