The moduli space of stable real cubic surfaces is the quotient of real
hyperbolic four-space by a discrete, nonarithmetic group. The volume of the
moduli space is 37\pi^2/1080 in the metric of constant curvature -1. Each of
the five connected components of the moduli space can be described as the
quotient of real hyperbolic four-space by a specific arithmetic group. We
compute the volumes of these components.
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