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Abstract

Locally compact connected topological non-Desarguesian translation planes have been a popular subject for research in geometry since the seventies of the last century. These planes are coordinatized by locally compact quasifields $(Q,+, \cdot )$ such that the kernel of $Q$ is either the field $\mathbb R$ of real numbers or the field $\mathbb C$ of complex numbers. In recent papers we determined the algebraic structure of the multiplicative loops $Q^{\ast }=(Q \setminus \{ 0 \}, \cdot )$ of quasifields $Q$ such that $Q$ has dimension $2$ over its kernel. Now we compare these cases and give a unified treatment of our results. In particular, we deal with multiplicative loops which either have a one-dimensional normal subloop or contain a compact subgroup.