Affine deformations of ultraideal triangle groups
type: Item
instance of: scholarly article
title: Affine deformations of ultraideal triangle groups
zbMATH Open document ID: 1041.57007
DOI: 10.1023/A:1023632111963
author: Virginie Charette
published in: Geometriae Dedicata
publication date: 2003-08-06T00:00:00Z
review text: 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.
reviewed by: Emil Molnár
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 1960040
zbMATH Keywords: proper actions of triangle groups
zbMATH Keywords: Lorentz manifolds
MaRDI profile type: Publication
full work available at URL: a:1023632111963
OpenAlex ID: W194373839
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description: scientific article; zbMATH DE number 1960040