  We show that for three dimensional gravity with higher genus boundary
conditions, if the theory possesses a sufficiently light scalar, there is a
second order phase transition where the scalar field condenses. This three
dimensional version of the holographic superconducting phase transition occurs
even though the pure gravity solutions are locally AdS$_3$. This is in addition
to the first order Hawking-Page-like phase transitions between different
locally AdS$_3$ handlebodies. This implies that the R\'enyi entropies of
holographic CFTs will undergo phase transitions as the R\'enyi parameter is
varied, as long as the theory possesses a scalar operator which is lighter than
a certain critical dimension. We show that this critical dimension has an
elegant mathematical interpretation as the Hausdorff dimension of the limit set
of a quotient group of AdS$_3$, and use this to compute it, analytically near
the boundary of moduli space and numerically in the interior of moduli space.
We compare this to a CFT computation generalizing recent work of Belin, Keller
and Zadeh, bounding the critical dimension using higher genus conformal blocks,
and find a surprisingly good match.
