Abstract
Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
Highlights
Résumé (Croissance des ensembles produit dans les groupes de Burnside) Étant donné un quotient périodique d’un groupe hyperbolique sans torsion, nous donnons une estimation inférieure fine de la fonction de croissance pour chacun de tous ses sous-semigroupes
If G is a free group, Safin [Saf11], improving former results by Chang [Cha08] and Razborov [Raz14], proves that there exists c > 0 such that for every finite subset V ⊂ G, either V is contained in a cyclic subgroup, or for every r ∈ N, we have
A similar statement holds for SL2(Z) [Cha08], free products, limit groups [But13] and groups acting on δ-hyperbolic spaces [DS20]
Summary
Burnside groups. — Let us explain the main idea behind the proof of Theorem 1.2. For simplicity we restrict ourselves to the case of free Burnside groups of rank 2. As we stressed before the constant C involved in Theorem 1.6 only depends on those parameters It holds, with the same constant C, for each group Gi acting on Xi. Consider a subset V ⊂ B2(n) that is not contained in a finite subgroup. We have a control of the length of every element in Uj. we have a control of the length of every element in Uj This allow us to lift Uj+1 to a finite subset Uj ⊂ Gj whose energy is controlled (i.e. bounded above by some C ). In particular one cannot control acylindricity parameters along the sequence (Gi), which means that our strategy fails here. We thank the referees for their careful reading and helpful comments
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