Frobenius Problem and Dead Ends in Integers

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. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In the appendix we show that every finitely generated group has a generating set with respect to which dead ends exist. Introduction We first describe the variant of Frobenius Problem that is in our interest. Definition 1 (Frobenius values). Let S be a set of positive integers, whose greatest common divisor is 1. An integer n is termed positively generated with respect to S if it is a non-negative integer linear combination of the elements in S, negatively generated if it is a non-positive integer linear combination of the elements in S, and is termed Frobenuis value (with respect to S) otherwise. It is known that for any set S of positive integers with greatest common divisor 1, there exist only finitely many Frobenius values. Frobenius Problem (also known as linear Diophantine problem of Frobenius) asks to find the largest Frobenius value for a given S. The largest Frobenius value is called the Frobenius number of S. It is known that, for S = {a, b}, where a > b > 1 are relatively prime, the Frobenius number is (a − 1)(b − 1) − 1. No explicit formula exists when S consists of at least three distinct numbers. On the positive side, upper bounds exist (see [BDR02] and [FR] for some estimates and further references) as do polynomial time algorithms determining the Frobenius number for sets S of fixed size [Kan92]. While Frobenius Problem has a long history, the notion of a dead end is fairly recent. It appears explicitly in the work of Bogopolskĭı [Bog97], who shows that for a given hyperbolic group with a given finite generating set, there exists a uniform bound on the depth of the dead ends in the group. Various results regarding dead ends in Thompson’s group F , lamplighter groups, solvable groups, finitely presented groups, residually finite groups, etc., appear in the works of Cleary, Guba, Riley, Taback and Warshall [CT04, CT05, Gub05, CR06, RW05, War06a, War06b]. Definition 2 (Word length). Let G be a group generated by a finite set S. For an element g in G, define the word length (or simply length) of g with respect to 2000 Mathematics Subject Classification. 11D04,05C25,20F65.