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On Weighted Geodesics in Groups
Published online by Cambridge University Press: 20 November 2018
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.
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- Copyright © Canadian Mathematical Society 01
References
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Rotman, J.J, The Theory of Groups: An Introduction, 2nd ed., Allyn and Bacon, Boston, 1973.Google Scholar
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