We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup H depending on the relative quasiconvexity of the intersection of H with the vertex groups of G. We give an application towards local relative quasiconvexity.

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

a b s t r a c tWe show that a group with a presentation satisfying the C(6) small-cancellation condition cannot contain a subgroup isomorphic to F 2 × F 2 .

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For any finitely generated, non-elementary, torsion-free group G that is hyperbolic relative to P, we show that there exists a group G * containing G such that G * is hyperbolic relative to P and G is not relatively quasiconvex in G * . This generalizes a result of I. Kapovich for hyperbolic groups. We also prove that any torsion-free group G that is non-elementary and hyperbolic relative to P, contains a rank 2 free subgroup F such that the group generated by "randomly" chosen elements r1, . . . , rm in F is aparabolic, malnormal in G and quasiconvex relative to P and therefore hyperbolically embedded relative to P.