Semihyperbolic Groups

Abstract

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 associated to some finite generating set it is a semihyperbolic metric space. Such a group is of type FPx and satisfies a quadratic isoperimetric inequality. We define a group to be semihyperbolic if it satisfies a stronger (equivariant) condition. We prove that this class of groups has strong closure properties. Word-hyperbolic groups and biautomatic groups are semihyperbolic. So too is any group which acts properly and cocompactly by isometries on a space of non-positive curvature. A discrete group of isometries of a 3-dimensional geometry is not semihyperbolic 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 extension of an abelian group to be semihyperbolic; we give sufficient conditions for more general extensions. Semihyperbolic groups have a solvable conjugacy problem. We prove an algebraic version of the flat torus theorem; this includes a proof that a polycyclic group is a subgroup of a semihyperbolic group if and only if it is virtually abelian. We answer a question of Gersten and Short concerning rational structures on Zn.