Any two non-cyclic point stabilizers in a hyperbolic-like circle action have an explicit proper ping-pong partition, so their generated subgroup is their free product.
Solodov,Topological problems in the theory of dynamical systems, Uspekhi Mat
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Ping-pong dynamics of hyperbolic-like actions with non-simple points
Any two non-cyclic point stabilizers in a hyperbolic-like circle action have an explicit proper ping-pong partition, so their generated subgroup is their free product.