Independent loxodromics for BV acting on the ray graph

Determine whether the action of the braided Thompson group BV on the ray graph R contains two independent loxodromic elements.

Background

The braided Thompson group BV, though not acylindrically hyperbolic, acts on the ray graph R, a Gromov-hyperbolic graph associated to the mapping class group of the plane minus a Cantor set. The authors show various kernel equalities for BV and discuss implications for acylindrical hyperbolicity.

They note that whether BV’s action on R has two independent loxodromic elements remains unresolved; answering this would clarify the relationship between their kernel equalities and acylindrical hyperbolicity in this setting.