GROUPS WITH CONTEXT-FREE REDUCED WORD PROBLEM

ABSTRACT

In this paper we consider reduced word problems of groups. We explain the relationship between the word problem and the reduced word problem, and we give necessary and sufficient conditions for a language to be the word problem (or the reduced word problem) of a group. In addition, we show that the reduced word problem is recursive (or recursively enumerable) precisely when the word problem is recursive.

We then prove that the groups which have context-free reduced word problem with respect to some finite monoid generating set are exactly the context-free groups, thus proving a conjecture of Haring-Smith.

We also show that, if a group G has finite irreducible word problem with respect to a monoid generating set X, then the reduced word problem of G with respect to X is simple; this is a generalization of one direction of a theorem of Haring-Smith.

ACKNOWLEDGMENTS

The authors would like to thank the referee for carefully reading the paper and making constructive suggestions. The second author would also like to thank Hilary Craig for all her help and encouragement.