No 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.

44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's commentsInternational audienceWe establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri typ

If 1 ; : : : ; n are limit groups and S 1 n is of type FP n ‫/ޑ.‬ then S contains a subgroup of finite index that is itself a direct product of at most n limit groups. This answers a question of Sela.