Which Words have an Efficient Result?

Abstract

Black [Black, S. (1999). Which words spell “almost nilpotent?” J. Algebra 221: 475–496] defined a class of words which are efficient or which have an efficient result. The variety of residually finite groups defined by a word of this class, like that defined by a positive word, is nilpotent-of-bounded-class-by-bounded-exponent. We introduce a notion of a standard exponent for a homogeneous word and we show that a homogeneous word has an efficient result if and only if its standard exponent is 1. With this we shall improve the main results of Black (1999) including its theorem. We also answer its first two open questions.

Key Words:

• Word
• Efficient result
• Standard exponent

Mathematics Subject Classification:

• 20F10
• 20E10

Acknowledgments

I am very grateful to the referee for pointing out inaccuracies and giving careful suggestions for improvement. I thank Professor Christian Michaux and Francoise Point for their careful reading of the paper and their valuable suggestions. I also thank Professor Maurice Boffa for his invitation such that I have a chance to work at the Logic Team of the Institute of Mathematics in the University of Mons-Hainaut. This work was supported by Shanxi Scholarship Council of China.

Notes

^#Communicated by H. D. Macpherson.