ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 14, 20 Benyash-Krivetz Valerii, Institute of Mathematics, Belarus On decomposing some finitely generated groups into amalgamated free products A decomposition of a group Γ as a non-trivial free product with amalgamation (an FPA decomposition) can often be used to obtain important properties of Γ. Let X_{2}^{S}(Γ) be the variety of characters of irreducible representations of Γ in SL_{2}(ℂ). Theorem 1. If dim X_{2}^{S}(Γ) ≥ 2 then Γ has an FPA decomposition. C. T. C. Wall posed the following question (1977): which one-relator groups have FPA decompositions? Baumslag and Shalen [1] proved that if such a group G has ≥ 3 generators then G has an FPA decomposition. If G is an one-relator two generator group with torsion then obviously dim X_{2}^{S}(G) ≥ 2 and by Theorem 1 G is an FPA group (which proves conjecture of Fine, Levin and Rosenberger [2]). Theorem 2. If G = ⟨a, b | a^{2n} = R^{m}(a, b) = 1⟩, where n ≥ 0, m ≥ 2 and R(a, b) is a cyclically reduced word involving b then G has an FPA decomposition. As a corollary we obtain that Fuchsian groups G_{1} = ⟨a, b | [a, b]^{n} = 1⟩, G_{2} = ⟨a, b | a^{2} = [a, b]^{n} = 1⟩ have FPA decompositions for any n ≥ 2. References: 1. G. Baumslag, P. Shalen, Amalgamated products and finitely presented groups, Comment. Math. Helv. 65(1990), 243–254. 2. B. Fine, F. Levin, G. Rosenberger, Free subgroups and decompositions of one-relator products of cyclics; Part 2: Normal torsion free subgroups and FPA decompositiomns, J. of Indian Math. Soc. 49(1985), 237–247.