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.

The k -dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k -spheres mapped into k -connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each nonnegative integer matrix P and positive rational number r , we associate a finite, aspherical 2-complex X r;P and determine the Dehn function of its fundamental group G r;P in terms of r and the Perron-Frobenius eigenvalue of P . The range of functions obtained includes ı.x/ D x s , where s 2 Q \ OE2; 1/ is arbitrary. Next, special features of the groups G r;P allow us to construct iterated multiple HNN extensions which exhibit similar isoperimetric behavior in higher dimensions. In particular, for each positive integer k and rational s > .k C 1/=k , there exists a group with k -dimensional Dehn function x s . Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs .M; @M / in addition to .B kC1 ; S k /.20F65; 20F69, 20E06, 57M07, 57M20, 53C99 IntroductionGiven a k -connected complex or manifold one wants to identify functions that bound the volume of efficient ball-fillings for spheres mapped into that space. The purpose of this article is to advance the understanding of which functions can arise when one seeks optimal bounds in the universal cover of a compact space. Despite the geometric nature of both the problem and its solutions, our initial impetus for studying isoperimetric problems comes from algebra, more specifically the word problem for groups.The quest to understand the complexity of word problems has been at the heart of combinatorial group theory since its inception. When one attacks the word problem for a finitely presented group G directly, the most natural measure of complexity is What Brady and Bridson actually do in [3] is associate to each pair of positive integers p > q a finite aspherical 2-complex whose fundamental group G p;q has Dehn function x 2 log 2 2p=q . These complexes are obtained by attaching a pair of annuli to a torus, the attaching maps being chosen so as to ensure the existence of a family of discs in the universal cover that display a certain snowflake geometry (cf Figure 4 below). In the present article we present a more sophisticated version of the snowflake construction that yields a much larger class of isoperimetric exponents.Theorem A Let P be an irreducible nonnegative integer matrix with Perron-Frobenius eigenvalue > 1, and let r be a rational number greater than every row sum of P . Then there is a finitely presented group G r;P with Dehn function ı.x/ ' x 2 log .r / .Here, ' denotes coarse Lipschitz equivalence of functions. By taking P to be the 1 1 matrix .2 2q / and r D 2 p (for integers p > 2q ) we obtain the Dehn function ı.x/ ' x p=q and deduce the following corollary.Corollary B Q \ .2; 1/ IP. For each positive integer k one has the k -dimensional isoperimetric spe...

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form Γ × Γ where Γ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if P is a finitely generated, residually finite group with P ∼ = Γ × Γ then there is an embedding P ֒→ Γ × Γ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for P .

Abstract. We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups P ⊂ G where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for P ⊂ G is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that π 1 X is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of π 1 X.

The closure of the set of isoperimetric exponents for finitely presented groups is {1} ∪ [2, ∞). For each pair of positive integers p ≥ q, one can construct groups with aspherical presentations for which the Dehn function is n 2α , where α = log 2 (2p/q).Understanding the complexity of word problems in groups has been one of the central themes of combinatorial group theory in the twentieth century since the pioneering work of Max Dehn. In recent years, following the influential work of Mikhael Gromov [Gr1], attention has focused on estimating the complexity of the word problem by means of isoperimetric inequalities. This approach is based on the close connection between word problems in finitely presented groups and Plateau's problem concerning the filling of loops in Riemannian manifolds.A natural approach to the filling problem for loops in the universal cover of a closed Riemannian manifold M is to seek isoperimetric inequalities giving upper bounds on the area of discs with specified boundary and minimal area. The bounds are given as a function of the length of the boundary loop, and the function [0, ∞) → [0, ∞) giving the optimal bound is called F ill M 0 . Analogously, one can measure the complexity of the word problem in finitely presented groups by seeking isoperimetric inequalities giving upper bounds on the number of relators which one must apply in order to show that a word w in the given generators represents the trivial element in the group. The bounds are given in terms of the length of w, and the function N → N describing the optimal bound is called the Dehn function of the presentation.The Dehn functions associated to different presentations of a group are equivalent under the relation generated by composing functions with