SEMIHYPERBOLIC GROUPS

We define semihyperbolicity, a condition which describes non-positive curvature in the large for an arbitrary metric space. This property is invariant under quasi-isometry. A finitely generated group is said to be weakly semihyperbolic if when endowed with the word metric associa ted to some finite generating set it is a semihyperbolic metric space. Such a group is of type FP, and satisfies a quadratic isoperimetric i nequality. We define a group to be semihyperbolic if it satisfies a st ronger (equivariant) condition. We prove that this class of groups has strong closure properties. Word-hyperbolic groups and biautomatic gro ups are semihyperbolic. So too is any group which acts properly and co compactly by isometries on a space of non-positive curvature. A discre te group of isometries of a 3-dimensional geometry is not semihyperbol ic if and only if the geometry is Nil or Sol and the quotient orbifold is compact. We give necessary and sufficient conditions for a split e xtension of an abelian group to be semihyperbolic; we give sufficient conditions for more general extensions. Semihyperbolic groups have a s olvable conjugacy problem. We prove an algebraic version of the flat t orus theorem; this includes a proof that a polycyclic group is a subgr oup of a semihyperbolic group if and only if it is virtually abelian. We answer a question of Gersten and Short concerning rational structur es on Z(n).