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

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


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


Zbl 0642.57002
Full Text: DOI EuDML


[1] Alperin, R., Bass, H.: Length functions of group actions on ?-trees. Gersten, S.M., Stallings, J.R. (eds.). Combinatorial group theory and topology. Ann. Math. Stud.111 (1986) · Zbl 0978.20500
[2] Bergman, G.M.: A weak Nullstellensatz for valuations. Proc. Am. Math. Soc.28, 32-38 (1971) · Zbl 0193.00303
[3] Bieri, R., Groves, J.R.J.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math.347, 168-195 (1984) · Zbl 0526.13003
[4] Bieri, R., Strebel, R.: Almost finitely presented soluble groups. Comment. Math. Helv.53, 258-278 (1978) · Zbl 0373.20035
[5] Bieri, R., Strebel, R.: Valuations and finitely presented metabelian groups. Proc. Lond. Math. Soc. (3)41, 439-464 (1980) · Zbl 0448.20029
[6] Bieri, R., Strebel, R.: A geometric invariant for modules over an abelian group, J. Reine Angew. Math.322, 170-189 (1981) · Zbl 0448.20010
[7] Bieri, R., Neumann, W.D., Strebel, R.: A geometric invariant of discrete groups. Invent. Math.90, 451-477 (1987) · Zbl 0642.57002
[8] Bourbaki, N.: Commutative algebra. Hermann (Paris) and Addison-Wesley (Reading) 1972
[9] Brown, K.S.: Finiteness properties of groups. J. Pure Appl. Algebra44, 45-75 (1987) · Zbl 0613.20033
[10] Culler, M., Morgan, J.W.: Group actions onR-trees (preprint) · Zbl 0658.20021
[11] Houghton, C.H.: The first cohomology of a group with permutation module coefficients. Arch. Math.31, 254-258 (1978/1979) · Zbl 0377.20044
[12] Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Berlin Heidelberg New York: Springer 1977 · Zbl 0368.20023
[13] Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. 2nd edition, New York: Dover 1976 · Zbl 0362.20023
[14] Serre, J.-P.: Trees, Berlin Heidelberg New York: Springer 1980 (Translation of Arbres, amalgames, SL2, Astérisque 46 (1977))
[15] Tits, J.: A ?theorem of Lie-Kolchin? for trees. In: Contributions to algebra: A collection of papers dedicated to Ellis Kolchin, Academic Press, New York, 1977, pp. 377-388
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