Trees, valuations, and the Bieri-Neumann-Strebel invariant. (English) Zbl 0663.20033

Brown, Kenneth S.

Invent. Math. 90, 479-504 (1987).

The author introduces and studies HNN valuations on groups. These are functions v: \(G\to {\mathbb{R}}\cup \{\infty \}\) which satisfy axioms resembling those for valuations on rings. He relates them to decompositions of G as HNN-extensions. These valuations arise naturally in the classification of actions on \({\mathbb{R}}\)-trees whose hyperbolic length function is abelian. The author uses the HNN valuations to characterize the geometric invariant \(\Sigma\) of R. Bieri, W. D. Neumann and R. Strebel [ibid. 90, 451-477 (1987; Zbl 0642.57002)]. This turns out to be useful for computations, as the author illustrates by examples. In particular, he computes \(\Sigma\) for one relator groups G, which yields a description of the set of finitely generated normal subgroups of G with abelian quotients.

Reviewer: H.Abels

MSC:

20F05	Generators, relations, and presentations of groups
20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65	Geometric group theory
20J05	Homological methods in group theory
57M05	Fundamental group, presentations, free differential calculus

Keywords:

Bieri-Neumann-Strebel invariant; HNN valuations on groups; HNN- extensions; actions on \({\mathbb{R}}\)-trees; hyperbolic length function; geometric invariant; one relator groups; finitely generated normal subgroups

Citations:

Zbl 0642.57002

References:

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