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# Texas A&M Number Theory Seminar

##
Department of Mathematics

Milner 317

Wednesdays, 12:30-1:30 PM

Zoran Sunik

Texas A&M University
**Wednesday, February 14**

Milner 317, 12:30 PM

**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.