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Published online by Cambridge University Press: 05 April 2013
Problem 1 [I.M. Chiswell] A group G is n-residually free if, given any n non-trivial elements g1,…, gn of G, there exists a homomorphism ϕ: G → F to a free group F such that ϕ(gi) ≠ 1 for i = 1, …, n. A group G is fully residually free if it is n-residually free for all n. Finitely generated surface groups are fully residually free. Does there exist a finitely generated fully residually free group which is not finitely presented?
Problem 2 [I.M. Chiswell] Let Λ be a totally ordered abelian group. A group G is Λ-free if it has a free action on some Λ-tree, and G is tree-free if it is Λ-free for some Λ. If G is a finitely generated tree-free group, does G act freely on some AΛ-tree with A finitely generated?
This is true if G is fully residually free (see problem 1) (Remeslennikov). Not all tree-free groups are fully residually free: there are counterexamples due to D. Spellman.
Problem 3 [D.E. Cohen] For what meanings of “nice” is a graph product of nice groups nice?
Hermiller has shown that one “nice” property is that of having a finite, complete rewriting system. Baik, Howie and Pride (J. Algebra 162 (1993) 168-77) show that FP3 is a “nice” property, whilst Harlander and Meinert have shown that FPm is a “nice” property.
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