Texas A&M Number Theory Seminar

Abstract: We show that every finitely generated infinite group has a generating set with respect to which dead ends exist. We then pay particular attention to the most elementary case, namely the infinite cyclic group Z.

Let a and b be positive, relatively prime integers. We show that the following are equivalent:
    (i)  d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b,
    (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer
          linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property.

In addition, for given integers a and b, we explicitly describe all such elements in Z.