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