For a tame supercuspidal representation
π
\pi
of a connected reductive
p
p
-adic group
G
G
, we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of
G
G
, for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of
G
G
which is not inertially equivalent to
π
\pi
. The consequence is a set of broadly applicable tools for addressing the branching rules of
π
\pi
and the unicity of
[
G
,
π
]
G
[G,\pi ]_G
-types.