Fixed-point-free automorphism

Proot The universal cover of such a polycycle is an (r, r)-polycycle with a nontrivial group of fixed-point-free automorphisms. The list of (r, < )-polycycles for r,q < 4 is known (see Section 4.2). Inspection of this list gives only the infinite polycycles Prisntoo = P2X Pz and APrisrrioo = Tr%. Their orientable quotients are the infinite series of prisms Prismm (m > 2), the infinite series of antiprisms APrisnim... 

Let us now determine all elementary ( 2,3,4,5), 3)gOT-polycycles. Hie universal cover P of such a polycycle P is an elementary ( 2,3,4,5, 3)-polycycle, whichhas a non-trivial fixed-point-free automorphism group in Aut(P). Consideration of the above list of polycycles yields snub Prisma, as the only possibility. The polycycles snub Prismm and its non-orientable quotients arise in this process. ... 

If P is an elementary ( 2,3], 5)ge -polycycle, which is not a ( 2,3, 5)-polycycle, then its universal cover P is an elementary ( 2,3, 5)-polycycle, which has a fixed-point-free automorphism group included in Aut(P). Clearly, only snub APrismoo is such and it yields the infinite series of snub APrisnim and its non-orientable quotients. ... 

Proof. Hike such a polycycle. Its universal cover is an elementary (5,3)-polycycle, whose automorphism group contains some fixed-point-free transformation. Inspection of the list of elementary (5,3)-polycycles in Figure 7.2 yields only Eg = snub Prismoo as a possibility. Snub Prismm is obtained from the group of translations by m faces and the non-orientable quotients if the group contains also some translation followed by reflection. ... 

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