Words and Almost Nilpotent Varieties of Groups^#

^#Communicated by D. Macpherson.

For a word of a free group of rank n , the author obtains an invariant called its standard exponent, and shows that if any residually finite group satisfying the law defined by such a word is almost nilpotent, then the standard exponent of the word equals 1 .

Conversely, if the standard exponent of a word ω is 1 , then any residually finite or soluble group and any locally finite or soluble group satisfying the group law ω≡ 1 is nilpotent-of-bounded-class-by-bounded-exponent.

Keywords:

• Almost nilpotent
• Standard exponent
• Variety
• Word

2000 MR Subject Classification:

• 20F10
• 20E10

ACKNOWLEDGMENTS

I am grateful to the referee for his help on the English and for some revisions, in particular, his suggestions leading to simplifications of the proof of Theorem 23. I thank Professor Christian Michaux and Professor Francoise Point for their careful reading of the article and their valuable suggestions. This work was supported by Shanxi Scholarship Council of China.

Notes

^#Communicated by D. Macpherson.