Reidemeister classes in some weakly branch groups

TL;DR

This paper proves that certain classes of weakly branch groups have the property that all their automorphisms have infinite Reidemeister number, under specific conditions related to group structure and automorphism actions.

Contribution

It establishes the property $R_ty$ for saturated weakly branch groups under various structural and automorphism-related conditions, extending understanding of Reidemeister classes in these groups.

Findings

Weakly branch groups have $R_ty$ under specific automorphism conditions.

Automorphisms with finite order in $Out(G)$ imply $R_ty$.

Finitely generated, prime-branching, weakly stabilizer transitive groups also have $R_ty$.

Abstract

We prove that a saturated weakly branch group $G$ has the property $R_\infty$ (any automorphism $\phi:G\to G$ has infinite Reidemeister number) in each of the following cases: 1) any element of $Out(G)$ has finite order; 2) for any $\phi$ the number of orbits on levels of the tree automorphism $t$ inducing $\phi$ 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.