Related papers: Small cancelation rings
In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding…
Using small cancellation for rotating families of groups, we construct new examples of aspherical polyhedra.
In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large…
We study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, i.e. coincides with the positive theory of a non-abelian free group.…
Small cancellation groups form an interesting class with many desirable properties. It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no…
We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an…
Walker's cancellation theorem says that if B+Z is isomorphic to C+Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the…
We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that…
This note is intended as an introduction to two previous works respectively by Dahmani, Guirardel, Osin, and by Cantat, Lamy. We give two proofs of a Small Cancellation Theorem for groups acting on a simplicial tree. We discuss the…
In this paper we prove that small cancellation rings under some natural restrictions are non-amenable and contain non-commutative free associative algebra.
We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…
We give a combinatorial proof of a theorem of Gromov, which extends the scope of small cancellation theory to group presentations arising from labelled graphs.
We generalize a version of small cancellation theory to the class of acylindrically hyperbolic groups. This class contains many groups which admit some natural action on a hyperbolic space, including non-elementary hyperbolic and relatively…
We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality. In particular, we prove that we can endow a definable group with a definable topology making…
We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of…
We obtain a partial classification of the finite groups $G$ for which the integral group ring $\mathbb{Z} G$ has projective cancellation, i.e. for which $P \oplus \mathbb{Z} G \cong Q \oplus \mathbb{Z} G$ implies $P \cong Q$ for projective…
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $\mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by…
We give a new proof of the main theorem in the theory of C(6) small cancellation complexes. We prove the fundamental theorem of cubical small cancellation theory for C(9) cubical small cancellation complexes.
This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the…
There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom…