Reidemeister numbers of saturated weakly branch groups

Reidemeister classes in some weakly branch groups

We prove that a saturated weakly branch group $G$ has the property $R_\infty$ (any automorphism $ϕ:G\to G$ has infinite Reidemeister number) in each of the following cases: 1) any element of $Out(G)$ has finite order; 2) for any $ϕ$ the number of orbits on levels of the tree automorphism $t$ inducing $ϕ$ is uniformly bounded and $G$ is weakly stabilizer transitive; 3) $G$ is finitely generated, prime-branching, and weakly stabilizer transitive with some non-abelian stabilizers (with no restrictions on automorphisms). Some related facts and generalizations are proved.