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On Weighted Geodesics in Groups

Published online by Cambridge University Press:  20 November 2018

Seymour Lipschutz
Seymour Lipschutz*
Affiliation:
Department of Mathematics Temple University Philadelphia, PA 19122
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Abstract

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A word W in a group G is a geodesic (weighted geodesic) if W has minimum length (minimum weight with respect to a generator weight function α) among all words equal to W. For finitely generated groups, the word problem is equivalent to the geodesic problem. We prove: (i) There exists a group G with solvable word problem, but unsolvable geodesic problem, (ii) There exists a group G with a solvable weighted geodesic problem with respect to one weight function α1, but unsolvable with respect to a second weight function α2. (iii) The (ordinary) geodesic problem and the free-product geodesic problem are independent.

Keywords

20 F 10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 1 , 01 March 1987 , pp. 86 - 91
Copyright
Copyright © Canadian Mathematical Society 01

References

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2. Magnus, W., Uber diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz), J. Reine Angew. Math. 163 (1930), pp. 141–165.Google Scholar
3. Rotman, J.J, The Theory of Groups: An Introduction, 2nd ed., Allyn and Bacon, Boston, 1973.Google Scholar