Commensurability between once-punctured torus groups and once-punctured Klein bottle groups

摘要

The once-punctured torus and the once-punctured Klein bottle are topo-logically commensurable, in the sense that both of them are doubly covered by the twice-punctured torus. In this paper, we give a condition for a faithful type-preserving PSL(2, C)-representation of the fundamental group of the once-punctured Klein bottle to be "commensurable" with that of the once-punctured torus. We also show that such a pair of PSL(2,C)-representations extend to a representation of the fundamental group of a common quotient orbifold. Finally, we give an application to the study of the Ford domains.