We start from the embedding of the Klein model of a hyperbolic plane H over a Euclidean field K in its direct motion group G=PSL_2(K) and of both in PG(3,K) . We present a geometric procedure to obtain loops which are related to suitable regular subsets of direct motions as transversals of the coset space G/D , where D is the subgroup of hyperbolic rotations fixing a given point o∈H . We investigate some properties of such loops and we determine their automorphism groups.
Pianta, S., Pasotti, S., Zizioli, E., A Geometric Environment for Building up Loops, <<RESULTS IN MATHEMATICS>>, 2015; (Vol. 68, no. 3-4): 415-426. [doi:10.1007/s00025-015-0449-z] [http://hdl.handle.net/10807/66178]
A Geometric Environment for Building up Loops
Pianta, Silvia;Pasotti, Stefano;Zizioli, Elena
2015
Abstract
We start from the embedding of the Klein model of a hyperbolic plane H over a Euclidean field K in its direct motion group G=PSL_2(K) and of both in PG(3,K) . We present a geometric procedure to obtain loops which are related to suitable regular subsets of direct motions as transversals of the coset space G/D , where D is the subgroup of hyperbolic rotations fixing a given point o∈H . We investigate some properties of such loops and we determine their automorphism groups.
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