Affine deformations of ultraideal triangle groups (Q1397894)

This paper surveys some results of the author's doctoral dissertation [University of Maryland (2000)]. A group generated by 3 line inversions (reflections) of the affine space \(A(R^{2,1})\) with Lorentzian scalar product of signature \((2,1)\) in its vector space \(R^{2,1}\) of directions and the index-two normal subgroup (the triangle group) generated by taking products of pairs of these inversions are studied. Here the 3 lines are space-like. Crooked planes may be placed on each of the lines. Then the crooked planes are disjoint, they bound a fundamental domain for the proper action of the group generated by the inversions. In particular, its index-two hyperbolic subgroup acts properly, too. The Drumm-Goldman inequality system for disjointness of crooked planes [\textit{T. A. Drumm} and \textit{W. M. Goldman}, Topology 38, 323--351 (1999; Zbl 0941.51029)] provides a criterion for a fundamental domain to exist. There is a contrast between the linear (as well as for hyperbolic plane group) and the affine case. The author mentions some interesting phenomena in this informative paper. She formulates some beliefs supported by computer experiments, but also reports on counterexamples.