Commentarii Mathematici Helvetici

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Volume 76, Issue 2, 2001, pp. 314–338
DOI: 10.1007/PL00000381

Published online: 2001-06-30

The residual finiteness of positive one-relator groups

Daniel T. Wise[1]

(1) McGill University, Montreal, Canada

It is proven that every positive one-relator group which satisfies the $ {\rm C}'({1\over6}) $ condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every $ {\rm C}'({1\over6}) $ positive one-relator group is residually finite. It is shown that positive one-relator groups are generically $ {\rm C}'({1\over6}) $ and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs.

Keywords: Residually finite, one-relator group, malnormal, small cancellation theory

Wise Daniel: The residual finiteness of positive one-relator groups. Comment. Math. Helv. 76 (2001), 314-338. doi: 10.1007/PL00000381