  Topological crystalline insulators (TCIs) possess metallic surface states
protected by crystalline symmetry, which are a versatile platform for exploring
topological phenomena and potential applications. However, progress in this
field has been hindered by the challenge to probe optical and transport
properties of the surface states owing to the presence of bulk carriers. Here
we report infrared (IR) reflectance measurements of a TCI, (001) oriented
$Pb_{1-x}Sn_{x}Se$ in zero and high magnetic fields. We demonstrate that the
far-IR conductivity is unexpectedly dominated by the surface states as a result
of their unique band structure and the consequent small IR penetration depth.
Moreover, our experiments yield a surface mobility of 40000 $cm^{2}/(Vs)$,
which is one of the highest reported values in topological materials,
suggesting the viability of surface-dominated conduction in thin TCI crystals.
These findings pave the way for exploring many exotic transport and optical
phenomena and applications predicted for TCIs.
