Extended parallelity in spine spaces and its geometry (Q860218)

Spine spaces have been introduced by the first author in [J. Geom. 72, No. 1--2, 172--187 (2001; Zbl 1003.51001)] and studied intensely in their papers [Abh. Math. Semin. Univ. Hamb. 72, 59--77 (2002; Zbl 1023.51001), Demonstr. Math. 36, No. 4, 957--969 (2003; Zbl 1057.51001)], and in their paper with \textit{M. Pankov} [Abh. Math. Semin. Univ. Hamb. 75, 21--50 (2005; Zbl 1085.51005)] (see last cited paper for a definition of spine spaces). The main problem for spine spaces is that of characterizing synthetically (in terms of incidence and parallelism) their geometry. As a step in this direction, the authors extend a certain notion of natural parallelism and study the geometry of that extended parallelism, including its horizon and group of dilatations.