Limit sets of relatively hyperbolic groups

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundaries.     

Local quasiconvexity of groups acting on small cancellation complexes

Given a group acting cellularly and cocompactly on a simply connected 2-complex, we provide a criterion establishing that all finitely generated subgroups have quasiconvex orbits. This work generalizes the “perimeter method”. As an application, we show that high-powered one-relator products A∗B/《rⁿ》 are coherent if A and B are coherent.     

Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have a uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.     

Rings associated with hyperbolic groups

Abstract We define quasiconvexity cone Qcone(τ) over an infinite hyperbolic (in the sense of Gromov) group τ as the set of conjugacy classes of infinite quasiconvex subgroups H?τ and show that the abelian group of Qcone(τ)-divisors, i.e. finite sums of points from Qcone(τ) with integer coefficients, can be equipped with a natural structure of commutative associative ring with identity. Euler characteristic can be considered as a rational-valued function on Qcone(τ). This approach gives another point of view on the strengthened form of Hanna Neumann's conjecture on the maximal rank of the intersection of two finitely generated subgroups of the free group on two generators.     

Sectional curvature,compact cores,and local quasiconvexity

We define a new notion of sectional curvature for 2-complexes, and describe a variety of examples with nonpositive or negative sectional curvature. The 2-complexes with nonpositive sectional curvature have coherent and locally indicable fundamental groups. The 2-complexes with negative sectional curvature have the compact core property for covers with finitely generated fundamental group. The fundamental groups of compact 2-complexes with metric negative sectional curvature have locally-quasiconvex fundamental groups.     

Coxeter groups are virtually special

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups.     

Local homology and dimensional full-valuedness

A natural criterion for dimensional full-valuedness of locally compact spaces with finitely generated local homology is given.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.     

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature     

Malnormality is undecidable in hyperbolic groups

In answer to a question of Myasnikov, we show that there exist hyperbolic groups for which there is no algorithm to decide which finitely generated subgroups are malnormal or quasiconvex. Bridson’s research is supported by an EPSRC Advanced Fellowship and Wise is supported in part by a grant from the NSF.     

Homological Finiteness Conditions for Groups,Monoids, and Algebras

Let ? be an additive category and 𝒞 a full subcategory with split idempotents, and closed under isomorphic images and finite direct sums. We give conditions on ? and 𝒞 implying that ? embeds into an abelian category, so that the objects of 𝒞 turn into injective objects. This construction generalizes the embedding of exactly definable categories into locally coherent categories, while the dual construction generalizes the embedding of finitely accessible categories into Grothendieck categories with a family of finitely generated projective generators. As applications, we characterize exactly definable categories through intrinsic properties and study those locally coherent categories whose fp-injective objects form a Grothendieck category.     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in $\text{PSL}\text{(2,}\text{R}\text{)}$) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Stable generalization of convex functions

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.     

Nonpositive sectional curvature for -complexes

We give a criterion for the nonpositive sectional curvature of -complexes. As a consequence, we show that certain -complexes have locally indicable, coherent and even locally quasiconvex fundamental groups.

     

Positively finitely related profinite groups

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.     

关于凝聚局部环的正则性

本文证明了极大理想m是有限生成的交换凝聚局部环（R,M）是正则的充分必要条件是m可以由一个正则R-序列生成,推广了文献[1]中相应的结论并给出了一个由正则凝聚局部环构造大量的非正则凝聚局部环的方法．     

Geodetic rays and fibers in periodic graphs

Using the notion of fibers, where two rays belong to the same fiber if and only if they lie within bounded Hausdorff‐distance of one another, we study how many fibers of a graph contain a geodetic ray and how many essentially distinct geodetic rays such “geodetic fibers” must contain. A complete answer is provided in the case of locally finite graphs that admit an almost transitive action by some infinite finitely generated, abelian group. Such graphs turn out to have either finitely many or uncountably many geodetic fibers. Furthermore, with finitely many possible exceptions, each of these fibers contains uncountably many geodetic rays. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 67–88, 2000     

On coherence of graph products of groups and Coxeter groups

Graph products of groups and Coxeter groups are defined via vertex-edge-labeled graphs. We show that if the graph has a special shape, then the corresponding group is coherent, i.e. every finitely generated subgroup is finitely presented.