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 nperiodic 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 nperiodic groups and the SQ-universality (in the class of n-periodic groups) of free Burnside groups.The small cancellation theorem. Let us present now a simplified version of our main result. We use the usual definition of the classical C ′ (λ)-condition [31, Chapter V]; see also Definition 2.1. Roughly speaking, this condition on a presentation requires that whenever two relators r = r ′ have a common subword u, then |u| < λ min{|r|, |r ′ |}.Theorem 1.6. Let p ∈ N * . There exists n p ∈ N such that for every odd integer n n p the following holds. Let G = S | R be a non-cyclic group given by a classical C ′ (1/6)-presentation. Assume that no p-th power of a word is a subword of an element of R, and no r ∈ R is a proper power.Our proof, in fact, yields the much more general Theorems 2.4 and 6.3, encompassing Gromov's graphical small cancellation theory, as well as classical and graphical small cancellation theory over free products. The philosophy is always similar to the one of Theorem 1.6: if the small cancellation presentation defines a non-elementary group, and if some restrictions on proper powers are satisfied, then some of the standard conclusions of small cancellation theory hold. For example, nperiodic graphical small cancellation produces infinite n-periodic groups with prescribed subgraphs in their Cayley graphs, and n-periodic free product small cancellation produces infinite n-periodic quotients of free products of n-periodic groups in which each of the generating free factors survives as subgroup.Remark 1.7. Even in the case that both S and R are finite, the statement of Theorem 1.6 is not covered by prior results. It is known [39,17] that, given a torsion-free Gromov hyperbolic group G, there exists n G ∈ N such that for all odd integers n n G , the quotient G/G n is infinite. Note here that n G depends on the specific group G and, in fact, given an exponent n ∈ N, the proof only applies to finitely many hyperbolic groups G of a given rank. In our result, on the other hand, the constant n p is independent of the presentation S | R . We shall see in the following how this eas...

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“…\end{equation*}It follows that $$\begin{equation*} \mathrm{diam}(A(a)^{+(100\delta +\lambda +\mu )}\cap o_{i,n}^{-1}o_{j,n}A(a)^{+(100\delta +\lambda +\mu )})\geqslant N^{\prime }. \end{equation*}$$By [11, Lemma 2.13], we have $$\begin{align*} &\Delta (a,{(o_{i,n}^{-1}o_{j,n})}a{(o_{i,n}^{-1}o_{j,n})}^{-1})\\ &\quad \geqslant \mathrm{diam}(A(a)^{+(100\delta +\lambda +\mu )}\cap o_{i,n}^{-1}o_{j,n}A(a)^{+(100\delta +\lambda +\mu )}) - (204\delta +2\lambda +2\mu ) \\ &\quad \geqslant N^{\prime } - (204\delta +2\lambda +2\mu ) =N. \end{align*}$$It follows from this inequality that the element …”