Related papers: Group rings whose group of units is hyperbolic

We survey the known results regarding the boundaries of word-hyperbolic groups.

We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct…

We classify groups G such that the unit group U(ZG) is hypercentral. In the second part we classify groups G whose modular group algebra has hyperbolic unit group V(KG).

In this paper we study hyperbolicty of the universal group $U(P)$ of a pregroup $P$. Given a finitely generated group $G$ and a pregroup $P$ such that $G \simeq U(P)$, we provide a particular set of axioms on $P$ which ensure that $G$ is…

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis…

We investigate the algebraic K- and L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.

We study residual properties of relatively hyperbolic groups. In particular, we show that if a group $G$ is non-elementary and hyperbolic relative to a collection of proper subgroups, then $G$ is SQ-universal.

In this note the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied.

Formal languages based on the multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups.

We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a…

We show that a relatively hyperbolic group either is virtually cyclic or has uniform exponential growth.

The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition.…

For a given divison algebra of the quaternions we construct two types of units: Pell units and Gauss units. If K is a rational quadratic extension and G is a finite group, we classify R and G, s.t., the unit group U(RG) of augmentation one…

Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated.

We describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group.

Let $G$ be a word hyperbolic group. We prove that the algebraic $K$-theory groups of $\dbZ [G]$, $K_n(\dbZ[G])$, have finite rank for all $n\in \dbZ$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic…

We give a dynamical characterization of acylindrically hyperbolic groups. As an application, we prove that non-elementary convergence groups are acylindrically hyperbolic.

We give solutions to several decision problems in word hyperbolic groups

We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We…