# Reidemeister Classes in Some Weakly Branch Groups

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## Abstract

We prove that a saturated weakly branch group *G* on an infinite spherically symmetric rooted tree *T* (i.e., a group which acts on *T* faithfully, level-transitively, with nontrivial rigid stabilizers of all vertices, and with a transitive action on sub-trees of some characteristic subgroups of all level stabilizers) has the property *R*_{∞} (any automorphism *ϕ*: *G* → *G* has infinite Reidemeister number) in each of the following cases: (1) any element of Out(*G*) is of finite order; (2) for any *ϕ*, the number of orbits on levels of the tree automorphism *t*, such that *ϕ*(*g*) = *tgt*^{−1}, is uniformly bounded and *G* is weakly stabilizer transitive, i.e., the intersection of stabilizers of all vertices of any level, except for successors of one vertex of the previous level, acts transitively on these successors; (3) *G* is finitely generated, has prime branching numbers, and is weakly stabilizer transitive with some non-Abelian quotients of stabilizers (with no restrictions on automorphisms). Some related facts and generalizations are proved.

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