A new class of kinematic spaces (Q1977515)

The present paper is inspired by \textit{M. Marchi} and \textit{E. Zizioli} [Ann. Discrete Math. 18, 601-615 (1983; Zbl 0509.51011), cited as [MZ]] -- in which a large class of kinematic spaces (in the sense of H. Karzel) is introduced -- and the authors propose a generalization of the initial conditions stated therein. Indeed, in [MZ] one proves that given the kinematic space \((G,{\mathcal G},+)\) and given \(U\leq\Aut (G,{\mathcal G},+)\), then the semidirect product \(P=G\rtimes U\) becomes, in turn, a kinematic space provided the following condition holds \[ (K)\qquad \forall a\in G,\;\forall \mu\in U \setminus \{id\}\exists_1x\in G:x-\mu(x)=a. \] We notice that \((K)\) entails that the nontrivial automorphisms are fixed-point free (cf. condition K2 in [MZ]) and that the centre of \(P\) is trivial. The authors introduce the following (weaker) conditions: \[ K':\qquad \forall\mu,\nu\in U\setminus \{id\}: \text{Fix} \mu= \text{Fix} \nu, \] and \[ K'':\qquad \forall\mu\in U\setminus \{id\} \text{ the map }\mu-id\text{ is surjective} \] \((K''\) is just (K2) of [MZ]). These new conditions ensure that \(P=G\rtimes U\) is again a kinematic space and are indeed weaker than \((K)\): the authors exhibit an example involving an infinite dimensional vector space and a suitable subgroup \(U\leq GL(V,K)\); then \(P:=V \rtimes U\) verifies \((K')\) and \((K'')\), but not \((K)\). They also prove that \(P\) admits no semidirect product type splitting fulfilling \((K)\), since the centre of \(P\) is non-trivial. The paper is concise but clear and readable.