  We study Besov capacities in a compact Ahlfors regular metric measure space
by means of hyperbolic fillings of the space. This approach is applicable even
if the space does not support any Poincar\'e inequalities. As an application of
the Besov capacity estimates we show that if a homeomorphism between two
Ahlfors regular metric measure spaces preserves, under some additional
assumptions, certain Besov classes, then the homeomorphism is necessarily a
quasisymmetric map.
