Real length functions in groups
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- by Nancy Harrison
- Trans. Amer. Math. Soc. 174 (1972), 77-106
- DOI: https://doi.org/10.1090/S0002-9947-1972-0308283-0
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Abstract:
This paper is a study of the structure of a group $G$ equipped with a ’length’ function from $G$ to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon’s work on groups with integer-valued functions. A real length function is a function which assigns to each $g \in G$ a nonnegative real number $|g|$ such that the following axioms are satisfied:
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$|x| < |xx|$ if $x \ne 1$.
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$|x| = 0$ if and only if $x = 1$.
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$|{x^{ - 1}}| = |x|$.
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$c(x,y) \geq 0$ where $c(x,y) = 1/2(|x| + |y| - |x y^{-1}|)$.
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$c(x,y) \geq m$ and $c(y,z) \geq m$ imply $c(x,z) \geq m$.
In this paper structure theorems are obtained for the cases when $G$ is abelian and when $G$ can be generated by two elements. We first prove that if $G$ is abelian, then $G$ is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of $G$. We prove that if $G$ can be generated by two elements, then $G$ is either free or abelian.
References
- Roger C. Lyndon, Length functions in groups, Math. Scand. 12 (1963), 209–234. MR 163947, DOI 10.7146/math.scand.a-10684
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 174 (1972), 77-106
- MSC: Primary 20F99
- DOI: https://doi.org/10.1090/S0002-9947-1972-0308283-0
- MathSciNet review: 0308283