  The Advanced Virgo detector uses two monolithic optical cavities at its
output port to suppress higher order modes and radio frequency sidebands from
the carrier light used for gravitational wave detection. These two cavities in
series form the output mode cleaner. We present a measured upper limit on the
length noise of these cavities that is consistent with the thermo-refractive
noise prediction of $8 \times 10^{-16}\,\textrm{m/Hz}^{1/2}$ at 15 Hz. The
cavity length is controlled using Peltier cells and piezo-electric actuators to
maintain resonance on the incoming light. A length lock precision of $3.5
\times 10^{-13}\,\textrm{m}$ is achieved. These two results are combined to
demonstrate that the broadband length noise of the output mode cleaner in the
10-60 Hz band is at least a factor 10 below other expected noise sources in the
Advanced Virgo detector design configuration.
