Linear conjugator length for iterated cyclic amalgams of hyperbolic groups

Establish that every iterated cyclic amalgam of hyperbolic groups has a linear conjugator length function.

Background

The paper proves that all iterated cyclic amalgams of finitely generated free groups have linear conjugator length functions. This result extends earlier work on conjugacy in amalgamated products and shows that maximality of the cyclic edge groups is not necessary in the free-group setting, even when the resulting amalgamated product is no longer hyperbolic.

The authors propose extending their methods from free groups to hyperbolic groups. They note that replacing freeness with torsion-free hyperbolicity in the relevant hypotheses might suffice for torsion-free hyperbolic groups, but that additional tools from hyperbolic group theory would be required; torsion introduces further considerations. Thus, the conjecture remains unresolved in the stated generality.