The geometry of crooked planes
In [Topology 31, No. 4, 677-683 (1992; Zbl 0773.57008)], the first author constructed non-compact fundamental polyhedra for subgroups \(\Gamma\) of \(\text{ SO}_0(2,1)\) acting properly discontinuously on the Euclidean 3-space \(E\). This construction is based on Poincaré's fundamental polyhedron method. The quotient \(E/\Gamma\) is called a Margulis space-time and the arising polyhedra are bounded by certain polyhedral hypersurfaces in \(E\), called crooked planes. The paper under review investigates the geometry of crooked planes by giving a description of the intersection of crooked planes. In order to formulate the main result, we have to introduce some notation. Let \(h(.,.)\) denote the Lorentzian scalar product and \(*\) the Lorentzian vector product. Let \(W^+ := \{ x \in {\mathbb R}^{2,1} |\;h(x,x) = 0,\;x_3 > 0\}\), \(P(v) = \{ x \in {\mathbb R}^{2,1} |\;h(v,x) = 0 \}\), and \(P^+(v) := \text{ cl}((W^+ \setminus {\mathbb R}_+v) * v)\) for \(v \in W^+\) nonzero. For any spacelike vector \(v\) there exist unique \(x^-(v), x^+(v) \in P(v) \cap W^+\) of length 1 such that \(h(v,x^-(v) * x^+(v)) > 0\). For \(p \in E\) and \(v \in {\mathbb R}^{2,1}\) spacelike set \(W^+(v,p) := p + P^+(x^+(v))\), \(W^-(v,p) := p + P^+(x^-(v))\), and \(S(v,p) := p + \{ x \in {\mathbb R}^{2,1} |\;h(v,x) = 0,\;h(x,x) \leq 0\}\). The piecewise linear submanifold \(W^+(v,p) \cup W^-(v,p) \cup S(v,p)\) is called a (positively oriented) crooked plane \(C(v,p)\) with vertex \(p\) and direction vector \(v\). The main theorem reads as follows. Let \(v_1, v_2\) be two consistently oriented, ultraparallel, unit-spacelike vectors and \(p_1,p_2 \in E\). The positively oriented crooked planes \(C(v_1,p_1)\) and \(C(v_2,p_2)\) are disjoint if and only if \(h(p_2 - p_1, v_1 * v_2) > |h(p_2 - p_1,v_2)|+ |h(p_2 - p_1, v_2)|\).
scientific article; zbMATH DE number 1307829