Engel elements in weakly branch groups. (English) Zbl 1452.20034
A branch group is a special kind of group acting on spherically homogeneous rooted trees.
This paper is devoted to the study of properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, it is proved that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all \(p\)-elements and that the group is virtually a \(p\)-group for some prime \(p\). It is also showed that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition.
This paper is devoted to the study of properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, it is proved that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all \(p\)-elements and that the group is virtually a \(p\)-group for some prime \(p\). It is also showed that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition.
Reviewer: Egle Bettio (Venezia)
MSC:
| 20F45 | Engel conditions |
| 20E08 | Groups acting on trees |
| 20F50 | Periodic groups; locally finite groups |
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