Test elements in solvable Baumslag-Solitar groups

摘要

In this paper, normal forms are established for group elements in the solvable Baumslag-Solitar groups to classify all test elements in these groups. These normal forms are used to identify two general types of endomorphisms and automorphisms are identified through these types. Test elements are then identified as elements whose total exponents on one of the generators is zero. Finally, we show that Turner's Retract Theorem does not hold for these groups by giving a specific counterexample.