arXiv subjects

E. Rips

Publications and source records attributed to E. Rips.

4 recordsLinked to original sources

arXiv2020

Small cancelation rings

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gr\"obner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. This is a short version of paper arXiv:2010.02836

math.RA↗

arXiv2020

Group-like small cancellation theory for 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 defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Gr\"{o}bner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.

math.RA↗

arXiv2018

Construction of a quotient ring of $\mathbb{Z}_2\mathcal{F}$ in which a binomial $1 + w$ is invertible using small cancellation methods

We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring $\mathbb{Z}_2\mathcal{F} / \mathcal{I}$, where $\mathbb{Z}_2\mathcal{F}$ is the group algebra of the free group $\mathcal{F}$ over the field $\mathbb{Z}_2$, and the ideal $\mathcal{I}$ is generated by a single trinomial $1 + v + vw$, where $v$ is a complicated word depending on $w$. In $\mathbb{Z}_2\mathcal{F} / \mathcal{I}$ we have $(1 + w)^{-1} = v$, so $1 + w$ becomes invertible. We construct an explicit linear basis of $\mathbb{Z}_2\mathcal{F} / \mathcal{I}$ (thus showing that $\mathbb{Z}_2\mathcal{F} / \mathcal{I}\neq 0$). This is the first step in constructing rings with exotic properties.

math.RA↗

arXiv1998

Isoperimetric Functions of Groups and Computational Complexity of the Word Problem

We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial isoperimetric function. The embedding can be chosen in such a way that $G$ has bounded distortion in $H$.

math.GR↗