In FPP on hyperbolic groups, the set of exceptional directions has strictly smaller Hausdorff dimension than the boundary and exists densely with multiple disjoint geodesics when boundary dimension exceeds one.
Hyperbolic groups with low-dimensional boundary
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Geodesic Trees and Exceptional Directions in FPP on Hyperbolic Groups
In FPP on hyperbolic groups, the set of exceptional directions has strictly smaller Hausdorff dimension than the boundary and exists densely with multiple disjoint geodesics when boundary dimension exceeds one.