Structure of small cancellation rings
Mathematics Research Reports, Volume 2 (2021) , pp. 1-14.

The theory of small cancellation groups is well known. In this paper we study the notion of the Group-like Small Cancellation Ring. 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 and 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 Gröbner 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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/mrr.6
Classification: 20F67,  16S15,  16Z05
Keywords: small cancellation ring, turn, multi-turn, defining relations in rings, small cancellation group, group algebra, filtration, tensor products, Dehn’s algorithm, greedy algorithm, Gröbner basis
@article{MRR_2021__2__1_0,
     author = {Agatha Atkarskaya and Alexei Kanel-Belov and Eugene Plotkin and Eliyahu Rips},
     title = {Structure of small cancellation rings},
     journal = {Mathematics Research Reports},
     pages = {1--14},
     publisher = {MathOA foundation},
     volume = {2},
     year = {2021},
     doi = {10.5802/mrr.6},
     language = {en},
     url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.6/}
}
Agatha Atkarskaya; Alexei Kanel-Belov; Eugene Plotkin; Eliyahu Rips. Structure of small cancellation rings. Mathematics Research Reports, Volume 2 (2021) , pp. 1-14. doi : 10.5802/mrr.6. https://mrr.centre-mersenne.org/articles/10.5802/mrr.6/

[1] S. I. Adian The Burnside problem and identities in groups, Springer-Verlag, Berlin, 1979 | MR 537580

[2] A. Atkarskaya; A. Kanel-Belov; E. Plotkin; E. Rips Construction of a quotient ring of 2 in which a binomial 1+w is invertible using small cancellation methods, Groups, Algebras, and Identities (Proc. of Israel Mathematical Conferences) (Contemporary Mathematics), Volume 726 (2019), pp. 1-76 | Article | MR 3937266 | Zbl 07119991

[3] A. Atkarskaya; A. Kanel-Belov; E. Plotkin; E. Rips Group-like small cancellation theory for rings (2020), 274 pages (arXiv:2010.02836)

[4] B. Bowditch A course on geometric group theory, 16, Mathematical Society of Japan, 2006 | Article | MR 2243589 | Zbl 1103.20037

[5] C. Druţu; M. Kapovich Geometric group theory, Colloquium Publications, Volume 63 (2018), p. 807 | MR 3753580

[6] M. Gromov Infinite groups as geometric objects, Proc. of Int. Congress Math. (1983), pp. 385-392

[7] M. Gromov Hyperbolic Groups, 8 (1987), pp. 75-263 | MR 919829 | Zbl 0634.20015

[8] V. Guba Finitely generated complete groups, Izv. Akad. Nauk SSSR Ser. Mat., Volume 50 (1986), pp. 883-924 | MR 873654

[9] V. Guba; M. Sapir Diagram groups, Memoirs of the American Mathematical Society, Volume 130 (1997) no. 620, pp. 1-117 | Article | MR 1396957 | Zbl 0930.20033

[10] P. M. Higgins Techniques of semigroup theory, Oxford University Press, Oxford, 1992 | Zbl 0744.20046

[11] S. Ivanov The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput., Volume 4 (1994), pp. 1-308 | Article | MR 1283947 | Zbl 0822.20044

[12] R. Lyndon On Dehn’s algorithm, Math. Ann., Volume 166 (1966), pp. 208-228 | Article | MR 214650 | Zbl 0138.25702

[13] R. Lyndon; P. Schupp Combinatorial group theory, Springer-Verlag, Berlin, 2001 (reprint of 1977 edition) | Article | Zbl 0997.20037

[14] I. Lysenok Infinite Burnside groups of even exponent, Izv. Math., Volume 60:3 (1996), pp. 453-654 | Article | MR 1405529 | Zbl 0926.20023

[15] P. S. Novikov; S. I. Adian Infinite periodic groups I, Math. USSR Izv., Volume 32 (1968), pp. 212-244 | MR 240178

[16] P. S. Novikov; S. I. Adian Infinite periodic groups II, Math. USSR Izv., Volume 32 (1968), pp. 251-524 | MR 240179

[17] P. S. Novikov; S. I. Adian Infinite periodic groups III, Math. USSR Izv., Volume 32 (1968), pp. 709-731 | MR 240180

[18] A. Olshanskii An infinite group with subgroups of prime orders, Math. USSR Izv., Volume 16 (1981), pp. 279-289 | Article

[19] A. Olshanskii Groups of bounded period with subgroups of prime order, Algebra and Logic, Volume 21 (1983), pp. 369-418 | Article

[20] A. Olshanskii Geometry of defining relations in groups, Mathematics and its Applications, Volume 70 (1991), p. 505 (translated from 1989 Russian original by Yu. A. Bakhturin) | MR 1191619

[21] E. Rips Generalized small cancellation theory and applications I, Israel J. Math., Volume 41 (1982), pp. 1-146 | Article | MR 657850 | Zbl 0508.20017

[22] M. Sapir Combinatorial algebra: syntax and semantics, Springer, Cham, 2014, 355 pages