Nonorientable Embeddings of Groups

Abstract

Embeddings of Cayley graphs into nonorientable surfaces are studied. Some lower and upper bounds for the nonorientable genus of abelian and hamiltonian groups are obtained. In the case when the lower and upper bounds coincide the nonorientable genus is computed. For example, let A = Z m 1 × Z m 2 × ⋯ × Z m r , where m i + 1 ∣ m i (1 ⩽ i ⩽ r - 1); if 4 divides m 1 , r ⩾ 3, and m r ⩾ 5, m r odd, then A has nonorientable genus 2 + ( r - 2)∣ A ∣/2. In some cases the method is applied to orientable embeddings. For example, if the abelian factor A of a hamiltonian group H has rank r ⩾ 6, then both the genus and the nonorientable genus of H are determined unless either ∣ A ∣ is odd or m r = 3.