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We exploit Zlil Sela's description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if G 1 ,. .. , G n are in E then a subgroup Γ ⊂ G 1 × · · · × G n is of type FP n if and only if Γ is itself, up to finite(More)
A subgroup of a product of n surface groups is of type F P n if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups. For John Stallings on his 65th birthday. By a surface group we mean the fundamental group of a connected 2-manifold. Such a group is either free (of finite or countably infinite rank) or(More)
We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of subdirect products of groups: if Γ 1 ,. .. , Γ n are finitely presented and S < Γ 1 ×· · ·×Γ n projects to a subgroup of finite index in each Γ i × Γ j , then S is finitely presentable, indeed there is an algorithm that will construct a finite presentation(More)
Our experience over 4 yr with direct questioning of patients reassures us that hypnosis and haemodynamic stability are provided reliably by the second method. We are currently investigating the dose, and evaluating the haemodynamic effects of propofol for induction and maintenance of hypnosis in the presence of full opioid analgesia in patients undergoing(More)
There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x −1 Ax = B for some x ∈ Γ with x less than a linear function of max{{γ : γ ∈ A ∪ B}. (The coefficients of this linear function depend(More)