Small cancellation theory over Burnside groups

Abstract

We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent n0 such that for every odd integer n⩾n0, the well-known classical C′(1/6)-small cancellation theory, as well as its graphical generalization and its version for free products, produce examples of infinite n-periodic groups. Our result gives a powerful tool for producing (uncountable collections of) examples of n-periodic groups with prescribed properties. It can be applied without any prior knowledge in the subject of n-periodic groups.As applications, we show the undecidability of Markov properties in classes of n-periodic groups, we produce n-periodic groups whose Cayley graph contains an embedded expander graphs, and we give an n-periodic version of the Rips construction. We also obtain simpler proofs of some known results like the existence of uncountably many finitely generated n-periodic groups and the SQ-universality (in the class of n-periodic groups) of free Burnside groups.